Inside Volatility Filtering: Secrets of the Skew [2nd ed.] 111894397X, 9781118943977

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Inside Volatility Filtering

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Inside Volatility Filtering Secrets of the Skew Second Edition

ALIREZA JAVAHERI

Copyright © 2015 by Alireza Javaheri. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. The First Edition of this book was published by John Wiley and Sons in 2005 under the title Inside Volatility Arbitrage: The Secrets of Skewness. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciically disclaim any implied warranties of merchantability or itness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993, or fax (317) 572-4002. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Cataloging-in-Publication Data: ISBN 978-1-118-94397-7 (cloth) ISBN 978-1-118-94399-1 (epdf) ISBN 978-1-118-94398-4 (epub) Cover Design: Wiley Cover Image: Alireza Javaheri Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

Foreword

ix

Acknowledgments (Second Edition)

xi

Acknowledgments (First Edition)

xiii

Introduction (Second Edition)

xv

Introduction (First Edition) Summary Contributions and Further Research Data and Programs

CHAPTER 1 The Volatility Problem Introduction The Stock Market The Stock Price Process Historic Volatility The Derivatives Market The Black-Scholes Approach The Cox Ross Rubinstein Approach Jump Diffusion and Level-Dependent Volatility Jump Diffusion Level-Dependent Volatility Local Volatility The Dupire Approach The Derman Kani Approach Stability Issues Calibration Frequency Stochastic Volatility Stochastic Volatility Processes GARCH and Diffusion Limits

xvii xvii xxiii xxiv

1 1 2 2 3 5 5 7 8 8 11 14 14 17 18 19 21 21 22

v

vi

CONTENTS

The Pricing PDE under Stochastic Volatility The Market Price of Volatility Risk The Two-Factor PDE The Generalized Fourier Transform The Transform Technique Special Cases The Mixing Solution The Romano Touzi Approach A One-Factor Monte-Carlo Technique The Long-Term Asymptotic Case The Deterministic Case The Stochastic Case A Series Expansion on Volatility-of-Volatility Local Volatility Stochastic Volatility Models Stochastic Implied Volatility Joint SPX and VIX Dynamics Pure-Jump Models Variance Gamma Variance Gamma with Stochastic Arrival Variance Gamma with Gamma Arrival Rate

CHAPTER 2 The Inference Problem Introduction Using Option Prices Conjugate Gradient (Fletcher-Reeves-Polak-Ribiere) Method Levenberg-Marquardt (LM) Method Direction Set (Powell) Method Numeric Tests The Distribution of the Errors Using Stock Prices The Likelihood Function Filtering The Simple and Extended Kalman Filters The Unscented Kalman Filter Kushner’s Nonlinear Filter Parameter Learning Parameter Estimation via MLE Diagnostics Particle Filtering Comparing Heston with Other Models

26 26 27 28 28 30 32 32 34 35 35 37 39 42 43 45 47 47 51 53

55 55 58 59 59 61 62 65 65 65 69 72 74 77 80 95 108 111 133

Contents

The Performance of the Inference Tools The Bayesian Approach Using the Characteristic Function Introducing Jumps Pure-Jump Models Recapitulation Model Identiication Convergence Issues and Solutions

CHAPTER 3 The Consistency Problem Introduction The Consistency Test The Setting The Cross-Sectional Results Time-Series Results Financial Interpretation The “Peso” Theory Background Numeric Results Trading Strategies Skewness Trades Kurtosis Trades Directional Risks An Exact Replication The Mirror Trades An Example of the Skewness Trade Multiple Trades High Volatility-of-Volatility and High Correlation Non-Gaussian Case VGSA A Word of Caution Foreign Exchange, Fixed Income, and Other Markets Foreign Exchange Fixed Income

CHAPTER 4 The Quality Problem Introduction An Exact Solution? Nonlinear Filtering Stochastic PDE

vii 141 158 172 174 184 201 201 202

203 203 206 206 206 209 210 214 214 215 216 216 217 217 219 220 220 225 225 230 232 236 237 237 238

241 241 241 242 243

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CONTENTS

Wiener Chaos Expansion First-Order WCE Simulations Second-Order WCE Quality of Observations Historic Spot Prices Historic Option Prices Conclusion

244 247 248 251 251 252 252 262

Bibliography

263

Index

279

Foreword revolution has been quietly brewing in quantitative inance for the last quarter century. In the 1980s and 1990s, quant groups were dominated by physicists and mathematicians well versed in sophisticated mathematical techniques related to physical phenomena such as heat low and diffusion. Rivers of data lowed through the canyons of Wall Street, but little thought was given to capturing the data for the purpose of extracting the signal from the noise. The dominant worldview at the time viewed inance as primarily a social enterprise, rendering it fundamentally distinct from the physical world that the quants had left behind. I think that most quants of the time would have agreed with George Bernard Shaw, who said that:

A

Hegel was right when he said that the only thing we learn from history is that man never learns anything from history. When an econometrician named Robert Litterman interviewed for a position at Goldman Sachs in 1986, the legendary Fischer Black began the interview by asking, “What makes you think that an econometrician has anything to contribute to Wall Street?” I suspect we will never know how the candidate answered that question, but had the candidate been Mark Twain, the answer would surely have been that: History doesn’t repeat itself, but it does rhyme. Six years after Black interviewed Litterman, they co-authored the Black– Litterman model that blends theoretical notions such as economic equilibrium with investor views guided by econometric time series models. Since then, econometricians and statisticians have enjoyed a steadily growing role in both practitioner and academic circles. In the May 2003 issue of Wilmott magazine, an article proiled past winners of the Nobel Memorial Prize in Economic Sciences. In a side note titled “A Winner You’d like to See,” the

The views represented herein are the author’s own views and do not necessarily represent the views of Morgan Stanley or its afliates, and are not a product of Morgan Stanley research.

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FOREWORD

author of this book provided the following quote about an NYU professor named Robert Engle: His work (with Bollerslev) on GARCH has made a real difference. Now, econometrics and quantitative inance are fully integrated thanks to him. . . . . I think the magazine could boldly suggest his nomination to Bank of Sweden! Half a year later, the Nobel committee appeared to have acted on Dr. Javaheri’s advice, awarding the Nobel Memorial Prize in Economic Sciences jointly to Professor Engle and another econometrician. The stochastic volatility models that represent the major focus of this book were speciically cited by the Nobel Committee as the basis for Professor Engle’s award. I believe that this watershed event changed the world of quantitative inance forever. Nowadays, few quants dispute the valuable insight that time series models bring to the dificult task of forecasting the future. At the Courant Institute of NYU where Dr. Javaheri and I co-teach a class, the master’s program in Mathematical Finance has one class each semester devoted to inancial econometrics. In particular, classes on machine learning and knowledge representation are routinely oversubscribed, alleviated only by the steady low of graduates to industry. The success of the statistical approach to artiicial intelligence has had a profound impact on inancial activity, leading to steadily increasing automation of both knowledge accumulation and securities trading. While e-trading typically starts with cash instruments and vanilla securities, it is inevitable that it will eventually encompass trading activities that lean heavily on quantitative elements such as volatility trading. As a result, the second edition of this book serves its intended audience well, providing an up-to-date, comprehensive review of the application of iltering techniques to volatility forecasting. While the title of each chapter is framed as a problem, the contents of each chapter represent our best guess at the answer. Employing the advances that econometricians have made in the past quarter century, the fraction of variance explained is a truly impressive accomplishment. Peter Carr Global Head of Market Modeling, Morgan Stanley Executive Director, Masters in Math Finance Program, Courant Institute, New York University

Acknowledgments (Second Edition)

’d like to thank the participants at the Global Derivatives 2012 and 2013 conferences for their valuable feedback. Many thanks go the students of the Baruch MFE program, where the irst edition of the book was used for a Volatility Filtering course. Special thanks go to Peter Carr for his unique insights, as usual. Finally, I’d like to once again thank my wife, Firoozeh, and my children, Neda, Ariana, Alexander, and Kouros, for their patience and support.

I

xi

Acknowledgments (First Edition)

his book is based on my PhD dissertation at Ecole des Mines de Paris. I would like to thank my advisor Alain Galli for his guidance and help. Many thanks to Margaret Armstrong and Delphine Lautier and the entire CERNA team for their support. Special thanks to Yves Rouchaleau for helping make all this possible in the irst place. I would like to sincerely thank other committee members Marco Avellaneda, Lane Hughston, Piotr Karasinski, and Bernard Lapeyre for their comments and time. I am grateful to Farshid Asl, Peter Carr, Raphael Douady, Robert Engle, Stephen Figlewski, Espen Haug, Ali Hirsa, Michael Johannes, Simon Julier, Alan Lewis, Dilip Madan, Vlad Piterbarg, David Wong, and the participants to ICBI 2003 and 2004 for all the interesting discussions and idea exchanges. I am particularly indebted to Paul Wilmott for encouraging me to speak to Wiley and convert my dissertation into this book. Finally, I’d like to thank my wife, Firoozeh, and my daughters, Neda and Ariana, for their patience and support.

T

xiii

Introduction (Second Edition)

he second edition of this book was written ten years after the irst one. Needless to say, a great number of things happened during this period. Many readers and students provided useful feedback leading to corrections and improvements. For one thing, various people correctly pointed out that the original title Inside Volatility Arbitrage was quite misleading since the term arbitrage was used in a loose statistical manner.1 Hence, the change in name to Inside Volatility Filtering. More to the point, the second edition beneited from various conference presentations I made, as well as courses I taught at the Courant Institute of Mathematics as well as at Baruch College. A few new topics are introduced and organized in the following way: The irst chapter continues to be a survey of literature on the topic of volatility. A few sections are added to go over innovations such as Local Vol Stochastic Vol models, models to represent SPX/VIX dynamics, and Stochastic Implied Vol. The second chapter still focuses on iltering and optimization. Two short sections on optimization techniques are added with a view to completeness. The third chapter tackles the consistency question. I tried to further stress that this chapter merely provides a methodology and does not contain a robust empirical work. As such, the examples in this chapter remain anecdotal. A fourth chapter is added to discuss some new ideas on the application of Wiener chaos to the estimation problem, as well as using better quality observations from options markets. A few new references are added. However, with the ever-increasing body of literature I am certain many have been missed.

T

1

As opposed to risk-less arbitrage in derivatives theory.

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Introduction (First Edition)

SUMMARY This book focuses on developing Methodologies for Estimating Stochastic Volatility (SV) parameters from the Stock-Price Time-Series under a Classical framework. The text contains three chapters and is structured as follows: In the irst chapter, we shall introduce and discuss the concept of various parametric SV models. This chapter represents a brief survey of the existing literature on the subject of nondeterministic volatility. We start with the concept of log-normal distribution and historic volatility. We then will introduce the Black-Scholes [40] framework. We shall also mention alternative interpretations as suggested by Cox and Rubinstein [71]. We shall state how these models are unable to explain the negative-skewness and the leptokurticity commonly observed in the stock markets. Also, the famous implied-volatility smile would not exist under these assumptions. At this point we consider the notion of level-dependent volatility as advanced by researchers such as Cox and Ross [69, 70] as well as Bensoussan, Crouhy, and Galai [34]. Either an artiicial expression of the instantaneous variance will be used, as is the case for Constant Elasticity Variance (CEV) models, or an implicit expression will be deduced from a Firm model similar to Merton’s [199], for instance. We also will bring up the subject of Poisson Jumps [200] in the distributions providing a negative-skewness and larger kurtosis. These jump-diffusion models offer a link between the volatility smile and credit phenomena. We then discuss the idea of Local Volatility [38] and its link to the instantaneous unobservable volatility. Work by researchers such as Dupire [94], Derman, and Kani [79] will be cited. We shall also describe the limitations of this idea due to an ill-poised inversion phenomenon, as revealed by Avellaneda [17] and others. Unlike Non-Parametric Local Volatility models, Parametric Stochastic Volatility (SV) models [147] deine a speciic stochastic differential equation for the unobservable instantaneous variance. We therefore will introduce the notion of two-factor Stochastic Volatility and its link to one-factor Generalized Auto-Regressive Conditionally Heteroskedastic (GARCH) processes [42]. The SV model class is the one we shall focus on. Studies by scholars such

xvii

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INTRODUCTION (FIRST EDITION)

as Engle [99], Nelson [204], and Heston [141] will be discussed at this juncture. We will briely mention related works on Stochastic Implied Volatility by Schonbucher [224], as well as Uncertain Volatility by Avellaneda [18]. Having introduced SV, we then discuss the two-factor Partial Differential Equations (PDE) and the incompleteness of the markets when only cash and the underlying asset are used for hedging. We then will examine Option Pricing techniques such as Inversion of the Fourier transform, Mixing Monte-Carlo, as well as a few asymptotic pricing techniques, as explained for instance by Lewis [185]. At this point we shall tackle the subject of pure-jump models such as Madan’s Variance Gamma [192] or its variants VG with Stochastic Arrivals (VGSA) [52]. The latter adds to the traditional VG a way to introduce the volatility clustering (persistence) phenomenon. We will mention the distribution of the stock market as well as various option pricing techniques under these models. The inversion of the characteristic function is clearly the method of choice for option pricing in this context. In the second chapter we will tackle the notion of Inference (or Parameter-Estimation) for Parametric SV models. We shall irst briely analyze the Cross-Sectional Inference and will then focus on the Time-Series Inference. We start with a concise description of cross-sectional estimation of SV parameters in a risk-neutral framework. A Least Squares Estimation (LSE) algorithm will be discussed. The Direction-Set optimization algorithm [214] will be also introduced at this point. The fact that this optimization algorithm does not use the gradient of the input-function is important, since we shall later deal with functions that contain jumps and are not necessarily differentiable everywhere. We then discuss the parameter inference from a Time-Series of the underlying asset in the real world. We shall do this in a Classical (Non-Bayesian) [252] framework and in particular we will estimate the parameters via a Maximization of Likelihood Estimation (MLE) [134] methodology. We shall explain the idea of MLE, its link to the Kullback-Leibler [105] distance, as well as the calculation of the Likelihood function for a two-factor SV model. We will see that unlike GARCH models, SV models do not admit an analytic (integrated) likelihood function. This is why we will need to introduce the concept of Filtering [136]. The idea behind Filtering is to obtain the best possible estimation of a hidden state given all the available information up to that point. This estimation is done in an iterative manner in two stages: The irst step is a Time Update where the prior distribution of the hidden state, at a given point in time, is determined from all the past information via a ChapmanKolmogorov equation. The second step would then involve a Measurement

Introduction (First Edition)

xix

Update where this prior distribution is used together with the conditional likelihood of the newest observation in order to compute the posterior distribution of the hidden state. The Bayes rule is used for this purpose. Once the posterior distribution is determined, it could be exploited for the optimal estimation of the hidden state. We shall start with the Gaussian case where the irst two moments characterize the entire distribution. For the Gaussian-Linear case, the optimal Kalman Filter (KF) [136] is introduced. Its nonlinear extension, the Extended KF (EKF), is described next. A more suitable version of KF for strongly nonlinear cases, the Unscented KF (UKF) [174], is also analyzed. In particular we will see how this ilter is related to Kushner’s Nonlinear Filter (NLF) [181, 182]. EKF uses a irst-order Taylor approximation upon the nonlinear transition and observation functions, in order to bring us back into a simple KF framework. On the other hand, UKF uses the true nonlinear functions without any approximation. It, however, supposes that the Gaussianity of the distribution is preserved through these functions. UKF determines the irst two moments via integrals that are computed on a few appropriately chosen “sigma points.” NLF does the same exact thing via a Gauss-Hermite quadrature. However NLF often introduces an extra centering step, which will avoid poor performance due to an insuficient intersection between the prior distribution and the conditional likelihood. As we shall observe, in addition to their use in the MLE approach, the Filters above could be applied to a direct estimation of the parameters via a Joint Filter (JF) [140]. The JF would simply involve the estimation of the parameters together with the hidden state via a dimension augmentation. In other words, one would treat the parameters as hidden states. After choosing initial conditions and applying the ilter to an observation data set, one would then disregard a number of initial points and take the average upon the remaining estimations. This initial rejected period is known as the “burn in” period. We will test various representations or State Space Models of the Stochastic Volatility models such as Heston’s [141]. The concept of Observability [215] will be introduced in this context. We will see that the parameter estimation is not always accurate given a limited amount of daily data. Before a closer analysis of the performance of these estimation methods, we shall introduce simulation-based Particle Filters (PF) [84, 128], which can be applied to non-Gaussian distributions. In a PF algorithm, the Importance Sampling technique is applied to the distribution. Points are simulated via a chosen proposal distribution and the resulting weights proportional to the conditional likelihood are computed. Since the variance

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INTRODUCTION (FIRST EDITION)

of these weights tends to increase over time and cause the algorithm to diverge, the simulated points go through a variance reduction technique commonly referred to as Resampling [15]. During this stage, points with too small a weight are disregarded, and points with large weights are reiterated. This technique could cause a Sample Impoverishment, which can be corrected via a Metropolis-Hastings Accept/Reject test. Work by researchers such as Doucet [84], Smith, and Gordon [128] are cited and used in this context. Needless to say, the choice of the proposal distribution could be fundamental in the success of the PF algorithm. The most natural choice would be to take a proposal distribution equal to the prior distribution of the hidden state. Even if this makes the computations simpler, the danger would be a non-alignment between the prior and the conditional likelihood as we previously mentioned. To avoid this, other proposal distributions taking into account the observation should be considered. The Extended PF (EPF) and the Unscented PF (UPF) [240] precisely do this by adding an extra Gaussian Filtering step to the process. Other techniques such as Auxiliary PF (APF) have been developed by Pitt and Shephard [213]. Interestingly, we will see that PF brings only marginal improvement to the traditional KF’s when applied to daily data. However, for a larger time-step where the nonlinearity is stronger, the PF does help more. At this point we also compare the Heston model to other SV models such as the “3/2” model [185] using real market data, and we will see that the latter performs better than the former. This is in line with the indings of Engle and Ishida [100]. We can therefore apply our inference tools to perform Model Identiication. Various Diagnostics [136] are used to judge the performance of the estimation tools. Mean Price Errors (MPE) and Root Mean Square Errors (RMSE) are calculated from the residual errors. The same residuals could be submitted to a Box-Ljung test, which will allow us to see whether they still contain auto-correlation. Other tests such as the Chi-Square Normality test as well as plots of Histograms and Variograms [115] are performed. Most importantly, for the Inference process, we backtest the tools upon artiicially simulated data, and we observe that although they give the correct answer asymptotically, the results remain inaccurate for a smaller amount of data points. It is reassuring to know that these observations are in agreement with work by other researchers such as Bagchi [20]. Here, we shall attempt to ind an explanation for this mediocre performance. One possible interpretation comes from the fact that in the SV problem, the parameters affect the noise of the observation and not its drift. This is doubly true of volatility-of-volatility and stock-volatility correlation, which affect the noise of the noise. We should, however, note that the product

Introduction (First Edition)

xxi

of these two parameters enters in the equations at the same level as the drift of the instantaneous variance, and it is precisely this product that appears in the skewness of the distribution. Indeed the instantaneous volatility is observable only at the second order of a Taylor (or Ito) expansion of the logarithm of the asset-price. This also explains why one-factor GARCH models do not have this issue. In their context the instantaneous volatility is perfectly known as a function of previous data points. The issue therefore seems to be a low Signal to Noise Ratio (SNR). We could improve our estimation by considering additional data points. Using a high frequency (several quotes a day) for the data does help in this context. However, one needs to obtain clean and reliable data irst. Also, we can see why a large time-step (e.g., yearly) makes the inference process more robust by improving the observation quality. Still, using a large time-step brings up other issues such as stronger nonlinearity as well as fewer available data points, not to mention the non-applicability of the Girasnov theorem. We shall analyze the sampling distributions of these parameters over many simulations and see how unbiased and eficient the estimators are. Not surprisingly, the ineficiency remains signiicant for a limited amount of data. One needs to question the performance of the actual Optimization algorithm as well. It is known that the greater the number of the parameters we are dealing with, the latter the Likelihood function and therefore the more dificult to ind a global optimum. Nevertheless, it is important to remember that the SNR and therefore the performance of the inference tool depend on the actual value of the parameters. Indeed, it is quite possible that the real parameters are such that the inference results are accurate. We then apply our PF to a jump-diffusion model (such as the Bates [29] model), and we will see that the estimation of the jump parameters is more robust than the estimation of the diffusion parameters. This reconirms that the estimation of parameters affecting the drift of the observation is more reliable. We inally apply the PF to non-Gaussian models such as VGSA [52], and we will observe similar results as for the diffusion-based models. Once again the VG parameters directly affecting the observation are easier to estimate, while the arrival rate parameters affecting the noise are more dificult to recover. Although, as mentioned, we use a Classical approach, we briely discuss Bayesian methods [36] such as Markov Chain Monte Carlo’s (MCMC) [171] including the Gibbs Sampler [60] and the Metropolis-Hastings (MH) [63]

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INTRODUCTION (FIRST EDITION)

algorithm. Bayesian methods consider the parameters not as ixed numbers, but random variables having a prior distribution. One then updates these distributions from the observations similarly to what is done in the Measurement Update step of a Filter. Sometimes the prior and posterior distributions of the parameters belong to the same family and are referred to as Conjugates. The parameters are inally estimated via an averaging procedure similar to the one employed in the JF. Whether the Bayesian methods are actually better or worse than the Classical ones has been a subject of long philosophical debate [252] and remains for the reader to decide. Other methodologies that differ from ours are the Non-Parametric (NP) and the Semi-Non-Parametric (SNP) ones. These methods are based on Kernel-based interpolation procedures, and have the obvious advantage of being less restrictive. However, Parametric models such as the ones used by us, offer the possibility of comparing and interpreting parameters such as drift and volatility of the instantaneous variance explicitly. Researchers such as Gallant, Tauchen [114], and Aït-Sahalia [7] use NP/SNP approaches. Finally, in the third chapter, we will apply the Parametric Inference methodologies to a few assets and will question the consistency of information contained in the options markets on the one hand, and in the stock market on the other hand. We shall see that there seems to be an excess negative-skewness and kurtosis in the former. This is in contradiction with the Gisanov theorem for a Heston model and could mean either that the model is misspeciied, or that there is a proitable transaction to be made. Another explanation could come from the Peso Theory [13] (or Crash-O-Phobia [162]) where an expectation of a so-far absent crash exists in the options markets. Adding a jump component to the distributions helps to reconcile the volatility of volatility and correlation parameters; however, it remains insuficient. This is in agreement with statements made by Bakshi, Cao, and Chen [21]. It is important to realize that ideally, one should compare the information embedded in the options and the evolution of the underlying asset during the life of these options. Indeed ordinary put or call options are forward (and not backward) looking. However given the limited amount of available daily data through this period, we make the assumption that the dynamics of the underlying asset do not change before and during the existence of the options. We therefore use time-series that start long before the commencement of these contracts. This assumption allows us to consider a Skewness Trade [7] where we would exploit such discrepancies by buying Out-of-the-Money (OTM) Call Options and selling OTM Put Options. We shall see that the results are not necessarily conclusive. Indeed, even if the trade often generates proits,

Introduction (First Edition)

xxiii

occasional sudden jumps cause large losses. This transaction is therefore similar to selling insurance. We also apply the same idea to the VGSA model where, despite the non-Gaussian features, the volatility of the arrival rate is supposed to be the same under the real and risk-neutral worlds. Let us be clear on the fact that this chapter does not constitute a thorough empirical study of stock versus options markets. It rather presents a set of examples of application for our previously constructed inference tools. There clearly could be many other applications, such as Model Identiication, as discussed in the second chapter. Yet another application of the separate estimations of the statistical and risk-neutral distributions is the determination of optimal positions in derivatives securities as discussed by Carr and Madan [56]. Indeed, the expected Utility function to be maximized needs the real-world distribution, while the initial wealth constraint exploits the risk-neutral distribution. This can be seen via a self-inancing portfolio argument similar to the one used by Black and Scholes [40]. Finally, we should remember that in all of these applications, we are assuming that the asset and options dynamics follow a known and ixed model such as Heston or VGSA. This is clearly a simpliication of reality. Indeed, the true markets follow an unknown and perhaps more importantly constantly changing model. The best we can do is to use the information hitherto available and hope that the future behavior of the assets is not too different from the past one. Needless to say, as time passes by and new information becomes available, we need to update our models and parameter values. This could be done within either a Bayesian or Classical framework. Also, we apply the same procedures to other asset classes such as Foreign-Exchange and Fixed-Income. It is noteworthy that although most of the text is centered on equities, almost no change whatsoever is necessary in order to apply the methodologies to these asset classes, which shows again how lexible the tools are. In the Bibliography section, many but not all relevant articles and books are cited. Only some of them are directly referred to in the text.

CONTRIBUTIONS AND FURTHER RESEARCH The contribution of the book is presenting a general and systematic way to calibrate any parametric SV model (diffusion-based or not) to a Time-Series under a Classical (non-Bayesian) framework. Although the concept of Filtering has been used for estimating volatility processes before [137], to our

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knowledge, this has always been for speciic cases and was never generalized. The use of Particle Filtering allows us to do this in a lexible and simple manner. We also studied the convergence properties of our tools and showed their limitations. Whether the results of these calibrations are consistent with the information contained in the options markets is a fundamental question. The applications of this test are numerous, among which the Skewness trade is only one example. What else can be done? A comparative study between our approach and Bayesian ones on the one hand, and Non-Parametric ones on the other hand. Work by researchers such as Johannes, Polson, and Aït-Sahalia would be extremely valuable in this context.

DATA AND PROGRAMS This book centers on Time-Series methodologies and exploits either artiicially generated inputs, or real market data. When real market data are utilized, the source is generally Bloomberg. However, most of the data could be obtained from other public sources available on the Internet. All numeric computations are performed via routines implemented in the C++ programming language. Some algorithms such as the Direction-Set Optimization algorithm are taken from “Numerical Recipes in C” [214]. No statistical packages such as S-Plus or R have been used. The actual C++ code for some of the crucial routines (such as EKF or UPF) is provided in the text.

Inside Volatility Filtering

CHAPTER

1

The Volatility Problem

Suppose we use the standard deviation of possible future returns on a stock as a measure of its volatility. Is it reasonable to take that volatility as a constant over time? I think not. —Fischer Black

INTRODUCTION It is widely accepted today that an assumption of a constant volatility fails to explain the existence of the volatility smile as well as the leptokurtic character (fat tails) of the stock distribution. The Fischer Black quote, made shortly after the famous constant-volatility Black-Scholes model was developed, proves the point. In this chapter, we will start by describing the concept of Brownian Motion for the Stock Price Return, as well as the concept of historic volatility. We will then discuss the derivatives market and the ideas of hedging and risk neutrality. We will briely describe the Black-Scholes Partial Derivatives Equation (PDE) in this section. Next, we will talk about jumps and level-dependent volatility models. We will irst mention the jump-diffusion process and introduce the concept of leverage. We will then refer to two popular level-dependent approaches: the Constant Elasticity Variance (CEV) model and the Bensoussan-CrouhyGalai (BCG) model. At this point, we will mention local volatility models developed in the recent past by Dupire and Derman-Kani and we will discuss their stability.

1

2

INSIDE VOLATILITY FILTERING

Following this, we will tackle the subject of stochastic volatility where we will mention a few popular models such as the Square-Root model and GARCH. We will then talk about the Pricing PDE under stochastic volatility and the risk-neutral version of it. For this we will need to introduce the concept of Market Price of Risk. The Generalized Fourier Transform is the subject of the following section. This technique was used by Alan Lewis extensively for solving stochastic volatility problems. Next, we will discuss the Mixing Solution, both in a correlated and noncorrelated case. We will mention its link to the Fundamental Transform and its usefulness for Monte-Carlo-based methods. We will then describe the Long-Term Asymptotic case, where we get closed-form approximations for many popular methods such as the Square-Root model. We will inally talk about pure-jump models such as Variance Gamma and VGSA.

THE STOCK MARKET The Stock Price Process The relationship between the stock market and the mathematical concept of Brownian Motion goes back to Bachelier [19]. A Brownian Motion corresponds to a process the increments of which are independent stationary normal random variables. Given that a Brownian Motion can take negative values, it cannot be used for the stock price. Instead, Samuelson [222] suggested to use this process to represent the return of the stock price, which will make the stock price a Geometric (or exponential) Brownian Motion. In other words, the stock price S follows a log-normal process1 dSt = �St dt + �St dBt

(1.1)

where dBt is a Brownian Motion process, � the instantaneous expected total return of the stock (possibly adjusted by a dividend yield), and � the instantaneous standard deviation of stock price returns, called the volatility in inancial markets. Using Ito’s lemma,2 we also have ( ) 1 2 d ln(St ) = � − � dt + �dBt . (1.2) 2 1 2

For an introduction to Stochastic Processes see Karatzas [175] or Oksendal [207]. See for example Hull [153].

3

The Volatility Problem

The stock return � could easily become time dependent without changing any of our arguments. For simplicity, we will often refer to it as � even if we mean �t . This remark holds for other quantities such as rt the interest-rate, or qt the dividend-yield. The equation (1.1) represents a continuous process. We can either take this as an approximation to the real discrete tick by tick stock movements, or consider it the real unobservable dynamics of the stock price, in which case the discrete prices constitute a sample from this continuous ideal process. Either way, the use of a continuous equation makes the pricing of inancial instruments more analytically tractable. The discrete equivalent of (1.2) is ln St+Δt

( ) √ 1 2 = ln St + � − � Δt + � ΔtBt 2

(1.3)

where Bt is a sequence of independent normal random variables with zero mean and variance of 1.

Historic Volatility This suggests a irst simple way to estimate the volatility �, namely the historic volatility. Considering S1 , . . . , SN a sequence of known historic daily stock close prices and calling Rn = ln(Sn+1 ∕Sn ) the stock price return between ∑N−1 two days and R = N1 n=0 Rn the mean return, the historic volatility would be the annualized standard deviation of the returns, namely

�hist

√ √ √ 252 N−1 ∑ (R − R)2 . =√ N − 1 n=0 n

(1.4)

Because we work with annualized quantities, and we are using daily stock close prices, we needed the factor 252, supposing that there are approximately 252 business days in a year.3 Note that N, the number of observations, can be more or less than one year; hence when talking about a historic volatility, it is important to know what time horizon we are considering. We can indeed have three-month historic volatility or three-year historic volatility. Needless to say, taking too few prices would give an inaccurate estimation. Similarly, the begin and end date of the observations matter. It is preferable to take the end date as close as possible to today, so that we include more recent observations.

3

Clearly the observation frequency does not have to be daily.

4

INSIDE VOLATILITY FILTERING

An alternative was suggested by Parkinson [210] where instead of daily close prices, we use the high and the low prices of the stock on that day, and high low Rn = ln(Sn ∕Sn ). The volatility would then be

�parkinson

√ √ N−1 √ 252 ∑ 1 (R − R)2 . =√ N − 1 4 ln(2) n=0 n

This second moment estimation derived by Parkinson is based on the fact that the range Rn of the asset follows a Feller distribution. Plotting for instance the one-year rolling4 historic volatility (1.4) of S&P 500 Stock Index, it is easily seen that this quantity is not constant over time. This observation was made as early as the 1960s by many inancial mathematicians and followers of the Chaos Theory. We therefore need time-varying volatility models. One natural extension of the constant volatility approach is to make �t a deterministic function of time. This is equivalent to giving the volatility a term structure, by analogy with interest rates. Historic Volatility 0.24 Historic Volatility

Historic Volatility

0.23 0.22 0.21 0.2 0.19 0.18

0

50

100

150 200 250 300 350 400 450 500 Days

SPX Historic Rolling Volatility from January 3, 2000, to December 31, 2001. As we can see, the volatility is clearly non-constant. 4

By rolling we mean that the one-year interval slides within the total observation period.

5

The Volatility Problem

THE DERIVATIVES MARKET Until now we only mentioned the stock price movements independently from the derivatives market. We now are going to include the inancial derivatives (specialty options) prices as well. These instruments became very popular and as liquid as the stocks themselves after Balck and Scholes introduced their risk-neutral pricing formula in [40].

The Black-Scholes Approach The Black-Scholes approach makes a number of reasonable assumptions about markets being frictionless and uses the log-normal model for the stock price movements. It also supposes a constant or deterministically time dependent stock drift and volatility. Under these conditions they prove that it is possible to hedge a position in a contingent claim dynamically by taking an offsetting position in the underlying stock and hence become immune to the stock movements. This risk neutrality is possible because, as they show, we can replicate the inancial derivative (for instance an option) by taking positions in cash and the underlying security. This condition of possibility of replication is called market completeness. In this situation everything happens as if we were replacing the stock drift �t with the risk-free rate of interest rt in (1.1), or rt − qt if there is a dividend-yield qt . The contingent claim f (S, t) having a payoff G(ST ) will satisfy the famous Black-Scholes equation rf =

�2 f �f �f 1 + (r − q)S + � 2 S2 2 . �t �S 2 �S

(1.5)

�f S is immune to the stock ranIndeed, the Hedged Portfolio Π = f − �S dom movements and according to Ito’s lemma veriies

( dΠ =

�f 1 2 2 � 2 f + � S �t 2 �S2

) dt

which must also be equal to rΠdt or else there would be possibility of Riskless Arbitrage.5 Note that this equation is closely related to the Feynman-Kac equation satisied by F(S, t) = Et (h(ST )) for any function h under the risk-neutral 5

For a detailed discussion see Hull [153].

6

INSIDE VOLATILITY FILTERING

measure. F(S, t) must be a martingale6 under this measure and therefore dBt and must be driftless, which implies dF = �S �F �S 0=

�F �F 1 2 2 � 2 F . + (r − q)S + � S �t �S 2 �S2

This would indeed be a different way to reach the same Black-Scholes equation, by using f (S, t) = exp(−rt)F(S, t), as was done for instance in Shreve [229]. Let us insist again on the fact that the real drift of the stock price does not appear in the previous equation, which makes the volatility �t the only unobservable quantity. As we said, the volatility could be a deterministic function of time without changing the earlier argument, in which case all we need to do is to t replace � 2 with 1t ∫0 �s2 ds and keep everything else the same. For calls and puts, where the payoffs G(ST ) are respectively MAX(0, ST − K) and MAX(0, K − ST ) where K is the strike price and T the maturity of the option, the Black Scholes Partial Derivatives Equation is solvable and gives the celebrated Black Scholes formula callt = St e−q(T−t) Φ(d1 ) − Ke−r(T−t) Φ(d2 )

(1.6)

putt = −St e−q(T−t) Φ(−d1 ) + Ke−r(T−t) Φ(−d2 )

(1.7)

and

where Φ(x) =

1 √ 2�

∫−∞ e− x

u2 2

du is the Cumulative Standard Normal function

√

and d1 = d2 + � T − t and d2 =

( ) ( ) S ln Kt + r−q− 21 � 2 (T−t) √ . � T−t

Note that using the well-known symmetry property for normal distributions Φ(−x) = 1 − Φ(x) in the above formulae, we could reach the Put-Call Parity relationship callt − putt = St e−q(T−t) − Ke−r(T−t)

(1.8)

that we can also rearrange as St e−q(T−t) − callt = Ke−r(T−t) − putt . The left-hand side of the last equation is called a covered call and is equivalent to a short position in a put combined with a bond. 6

For an explanation see Shreve [229] or Karatzas [175].

7

The Volatility Problem

The Cox Ross Rubinstein Approach Later, Cox, Ross, and Rubinstein [71] developed a simpliied approach using the Binomial Law to reach the same pricing formulae. The approach commonly referred to as the Binomial Tree uses a tree of recombining spot prices where at a given time-step n we have n + 1 possible S[n][j] spot prices, with 0 ≤ j ≤ n. Calling p the upward transition probability and 1 − p the downward transition probability, S the stock price today, and Su = uS and Sd = dS upper and lower possible future spot prices, we can write the expectation equation7 E[S] = puS + (1 − p)dS = erΔt S which immediately gives us p=

a−d u−d

with a = exp(rΔt). We can also write the variance equation Var[S] = pu2 S2 + (1 − p)d2 S2 − e2rΔt S2 ≈ � 2 S2 Δt which after choosing a centering condition such as ud = 1, will provide us √ √ with u = exp(� Δt) and d = exp(−� Δt). Using these values for u, d, and p, we can build the tree; and using the inal payoff, we can calculate the option price by backward induction.8 We can also build this tree by applying an Explicit Finite Difference scheme to the PDE (1.5) as was done in Wilmott [250]. An important advantage of the tree method is that it can be applied to American Options (with early exercise) as well. It is possible to deduce the implied volatility of call and put options by solving a reverse Black-Scholes equation: that is, ind the volatility that would equate the Black-Scholes price to the market price of the option. This is a good way to see how derivatives markets perceive the underlying volatility. It is easy to see that if we change the maturity and strike prices of options (and keep everything else ixed), the implied volatility will not be constant. It will have a linear skew and a convex form as the strike price

7 8

The expectation equation is written under the risk-neutral probability. For an in-depth discussion on Binomial Trees see Cox [72].

8

INSIDE VOLATILITY FILTERING

changes. This famous “smile” cannot be explained by simple time dependence, hence the necessity of introducing new models.9 Volatility Smile 0.32 0.3

Implied Volatility 1 month to Maturity Implied Volatility 7 months to Maturity

Implied Volatility

0.28 0.26 0.24 0.22 0.2 0.18 0.16 950

1000

1050

1100

1150

Strike Price

SPX volatility smile on February 12, 2002, with Index = 1107.5 USD, one month and seven months to maturity. The negative skewness is clearly visible. Note how the smile becomes latter as time to maturity increases.

JUMP DIFFUSION AND LEVEL-DEPENDENT VOLATILITY In addition to the volatility smile observable from the implied volatilities of the options, there is evidence that the assumption of a pure normal distribution (also called pure diffusion) for the stock return is not accurate. Indeed “fat tails” have been observed away from the mean of the stock return. This phenomenon is called leptokurticity and could be explained in many different ways.

Jump Diffusion Some try to explain the smile and the leptokurticity by changing the underlying stock distribution from a diffusion process to a jump-diffusion process.

9

It is interesting to note that this smile phenomenon was practically nonexistent prior to the 1987 stock market crash. Many researchers therefore believe that the markets have learned to factor in a crash possibility, which creates the volatility smile.

9

The Volatility Problem

A jump diffusion is not a level-dependent volatility process. However, we are mentioning it in this section to demonstrate the importance of the leverage effect. Merton [200] was irst to actually introduce jumps in the stock distribution. Kou [180] recently used the same idea to explain both the existence of fat tails and the volatility smile. The stock price will follow a modiied stochastic process under this assumption. If we add to the Brownian Motion dBt a Poisson (jump) process10 dq with an intensity11 �, then calling k = E(Y − 1) with Y − 1 the random variable percentage change in the stock price, we will have dSt = (� − �k)St dt + �St dBt + St dq

(1.9)

or equivalently ) ] [( �2 St = S0 exp � − − �k t + �Bt Yn 2 ∏ where Y0 = 1 and Yn = nj=1 Yj with Yj ’s independently identically distributed random variables and n a Poisson random variable with a parameter �t. It is worth noting that for the special case where the jump corresponds to total ruin or default, we have k = −1, which will give us dSt = (� + �)St dt + �St dBt + St dq and

[( St = S0 exp

�+�−

�2 2

)

(1.10)

] t + �Bt Yn .

Given that in this case E(Yn ) = E(Yn2 ) = e−�t it is fairly easy to see that in the risk-neutral world E(St ) = S0 ert exactly as in the pure diffusion case, but Var(St ) = S20 e2rt (e(�

2 +�)t

− 1) ≈ S20 (� 2 + �)t

(1.11)

unlike the pure diffusion case where Var(St ) ≈ S20 � 2 t. 10 11

See for instance Karatzas [175]. The intensity could be interpreted as the mean number of jumps per time unit.

10

INSIDE VOLATILITY FILTERING

Proof Indeed ( 2 ) � E(St ) = S0 exp((r + �)t) exp − t E[exp(�Bt )]E(Yn ) 2 ( 2 ) ( 2 ) � � = S0 exp((r + �)t) exp − t exp t exp(−�t) = S0 exp(rt) 2 2 and E(S2t ) = S20 exp(2(r + �)t) exp(−� 2 t)E[exp(2�Bt )]E(Yn2 ) ( ) (2�)2 = S20 exp(2(r + �)t) exp(−� 2 t) exp exp(−�t) 2 = S20 exp((2r + �)t) exp(� 2 t) and as usual Var(St ) = E(S2t ) − E2 (St ) (QED) Link to Credit Spread Note that for a zero-coupon risky bond Z with no recovery, a credit spread C and a face value X paid at time t we have Z = e−(r+C)t X = e−�t (e−rt X) + (1 − e−�t )(0); consequently, � = C, and using (1.8) we can write �̃ 2 (C) = � 2 + C where � is the ixed (pure diffusion) volatility and �̃ is the modiied jump diffusion volatility. This equation relates the volatility and leverage, a concept we will see later in level-dependent models as well. Also, we could see that everything happens as if we were using the Black-Scholes pricing equation but with a modiied “interest rate,” which �f is r + C. Indeed, the Hedged Portfolio Π = f − �S S now satisies ( dΠ =

�f 1 2 2 � 2 f + � S �t 2 �S2

) dt

under the no-default case, which occurs with a probability of e−�dt ≈ 1 − �dt and dΠ = −Π under the default case, which occurs with a probability of 1 − e−�dt ≈ �dt.

11

The Volatility Problem

We therefore have ( E(dΠ) =

) �f 1 2 2 � 2 f − �Π dt + � S �t 2 �S2

and using a diversiication argument we can always say that E(dΠ) = rΠdt, which provides us with (r + �)f =

�f �2f �f 1 + (r + �)S + � 2 S2 2 �t �S 2 �S

(1.12)

which again is the Black-Scholes PDE with a “risky rate.” A generalization of the jump-diffusion process would be the use of the Levy process, a stochastic process with independent and stationary increments. Both the Brownian Motion and the Poisson process are included in this category. For a description, see Matacz [196].

Level-Dependent Volatility Many assume that the smile and the fat tails are due to the level dependence of the volatility. The idea would be to make �t level dependent or a function of the spot itself; we would therefore have dSt = �t St dt + �(S, t)St dBt .

(1.13)

Note that to be exact, a level-dependent volatility is a function of the spot price alone. When the volatility is a function of the spot price and time, it is referred to as local volatility, which we shall discuss further. The Constant Elasticity Variance Approach One of the very irst attempts to use this approach was the Constant Elasticity Variance (CEV) method realized by Cox [69] and [70]. In this method, we would suppose an equation of the type (1.14) �(S, t) = CS�t where C and � are parameters to be calibrated either from the stock price returns themselves or from the option prices and their implied volatilities. The CEV method was recently analyzed by Jones [173] in a paper where he uses two � exponents. This level-dependent volatility represents an important feature that is observed in options markets as well as in the underlying prices: the negative correlation between the stock price and the volatility, also called the leverage effect.

12

INSIDE VOLATILITY FILTERING

The Bensoussan Crouhy Galai Approach Bensoussan, Crouhy, and Galai (BCG) [34] try to ind the level dependence of the volatility differently from Cox and Ross. Indeed, in the CEV model, Cox and Ross irst suppose that �(S, t) has a certain exponential form and only then try to calibrate the model parameters to the market. BCG, on the other hand, try to deduce the functional form of �(S, t) by using a irm structure model. The idea of irm structure is not new and goes back to Merton [199] where he considers that the Firm Assets follow a log-normal process dV = �V Vdt + �V VdBt

(1.15)

where �V and �V are the assets return and volatility. One important point is that �V is considered constant. Merton then argues that the equity S of the irm could be considered a call option on the assets of the irm with a strike price K equal to the face value of the irm liabilities and an expiration T equal to the average liability maturity. Using Ito’s lemma, it is fairly easy to see that ( ) �S �S �S 1 2 2 �2S dt + �V V + �V V + � V dB dS = �Sdt + �(S, t)SdBt = �t �V 2 V �V 2 �V t (1.16) which immediately provides us with �(S, t) = �V

V �S S �V

(1.17)

which is an implicit functional form for �(S, t). BCG then eliminate the asset term in the above functional form and end up with a nonlinear PDE �� �� 1 2 2 � 2 � + (r + � 2 )S + � S = 0. �t 2 �S2 �S

(1.18)

This PDE gives the dependence of � on S and t. Proof A quick sketch of the proof is as follows: S being a contingent-claim on V we have the risk-neutral Black-Scholes PDE �S 1 �2S �S + rV + �V2 V 2 2 = rS �t �V 2 �V and using

�S �V

= 1∕ �V as well as �S

�S �t

�S �V = − �V and �t

�2 S �V 2

have the reciprocal Black-Scholes equation �V 1 2 2 � 2 V �V + rS + � S = rV. �t �S 2 �S2

2

= − ��SV2

/(

�V �S

)3

we

13

The Volatility Problem

Now posing Ψ(S, t) = ln V(S, t), we have �V = V �Ψ as well as �t �t ) ( ) ( 2 2 2 and will have the new PDE and ��SV2 = V ��SΨ2 + �Ψ �S ( ( )2 ) �Ψ 1 2 2 � 2 Ψ �Ψ �Ψ + r= + rS + � S �t �S 2 �S2 �S and the equation

) /( �Ψ � = �V S . �S

This last identity implies the PDE becomes r=

�Ψ �S

=

�V S�

as well as

�2 Ψ �S2

=

( ) −�V �+S �� �S S2 � 2

�V �S

= V �Ψ �S

and therefore

( )) ( �� �Ψ 1 2 �V − �V � + S . + r�V ∕� + �t 2 �S 2

� Ψ Taking the derivative with respect to S and using �S�t = − S�V2 �� , we get �t the inal PDE �� �� 1 2 2 � 2 � + � S + (r + � 2 )S =0 �t 2 �S2 �S

as previously stated.

�

(QED)

We therefore have an implicit functional form for �(S, t) and just like for the CEV case, we need to calibrate the parameters to the market data. CEV Model 160 Market Model

140

Call Price

120 100 80 60 40 20 0 950

1000

1050

1100

1150

Strike Price

CEV model for SPX on February 12, 2002, with Index = 1107.5 USD, one month to maturity. The smile is itted well, but the model assumes a perfect (negative) correlation between the stock and the volatility.

14

INSIDE VOLATILITY FILTERING BCG Model 160 Market Model

140

Call Price

120 100 80 60 40 20 0 950

1000

1050 Strike Price

1100

1150

BCG model for SPX on February 12, 2002, with Index = 1107.5 USD, one month to maturity. The smile is itted well.

LOCAL VOLATILITY In the early 1990s, Dupire [94] on the one hand, and Derman & Kani [79] on the other, developed a concept called local volatility where the volatility smile was retrieved from the option prices.

The Dupire Approach The Breeden & Litzenberger Identity This approach uses the options prices to get the implied distribution for the underlying stock. To do this we can write V(S0 , K, T) = call(S0 , K, T) = e−rT

∫0

+∞

(S − K)+ p(S0 , S, T)dS

(1.19)

where S0 is the stock price at time t = 0 and K the strike price of the call, and p(S0 , S, T) the unknown transition density for the stock price. As usual, x+ = MAX(x, 0). Using the equation (1.19) and differentiating with respect to K twice, we get the Breeden & Litzenberger [47] implied distribution p(S0 , K, T) = erT

�2 V �K2

(1.20)

15

The Volatility Problem

Proof The proof is straightforward if we write ∫K

erT V(S0 , K, T) =

+∞

Sp(S0 , S, T)dS − K

∫K

+∞

p(S0 , S, T)dS

and take the irst derivative erT

�V = −Kp(S0 , K, T) + Kp(S0 , K, T) − ∫K �K

and the second derivative in the same manner.

+∞

p(S0 , S, T)dS

(QED)

The Dupire Identity Now according to the Fokker-Planck (or forward Kolmogorov) equation12 for this density, we have �p �(Sp) 1 � 2 (� 2 (S, t)S2 p) = −r �T 2 �S2 �S and therefore after a little rearrangement �V 1 �2V �V = � 2 K2 2 − rK �T 2 �K �K which provides us with the local volatility formula � 2 (K, T) =

�V + rK �V �T �K 1 2 �2 V K �K2 2

.

(1.21)

Proof For a quick proof of this, let us use the zero interest rates case (the general case could be done similarly). We would then have p(S0 , K, T) =

�2V �K2

as well as Fokker-Planck �p 1 � 2 (� 2 (S, t)S2 p) . = �T 2 �S2 12

See for example Wilmott [249] for an explanation on Fokker-Planck equation.

16

INSIDE VOLATILITY FILTERING

Now �V = �T ∫0

+∞

∫0

+∞

=

(ST − K)+

�p dS �T T

(ST − K)+

1 � 2 (� 2 (S, T)S2 p) dST 2 �S2

and integrating by parts twice and using the fact that � 2 (ST − K)+ = �(ST − K) �K2 with �(.) the Dirac function, we will have �V 1 1 �2 V = � 2 (K, T)K2 p(S0 , K, T) = K2 � 2 (K, T) 2 �T 2 2 �K as stated.

(QED)

It is also possible to use the implied volatility �BS from the Black-Scholes formula (1.5) and express the above local volatility in terms of �BS instead of V. For a detailed discussion, we could refer to Wilmott [249]. Local Volatility vs. Instantaneous Volatility Clearly the local volatility is related to the instantaneous variance vt , as Gatheral [118] shows. The relationship could be written as � 2 (K, T) = E[vT |ST = K]

(1.22)

That is, local variance is the risk-neutral expectation of the instantaneous variance conditional on the inal stock price being equal to the strike price.13 Proof Let us show the above identity for the case of zero interest rates.14 As mentioned above, we have 2

� (K, T) =

13

�V �T 1 2 �2 V K 2 �K2

.

Note that this is independent from the process for vt , meaning that any stochastic volatility model satisies this property, which is an attractive feature of local volatility models. 14 For the case of non-zero rates we need to work with the forward price instead of the stock price.

17

The Volatility Problem

On the other hand using the call payoff V(S0 , K, t = T) = E[(ST − K)+ ], we have �V = E[H(ST − K)] �K with H(.) the Heaviside function and �2V = E[�(ST − K)] �K2 with �(.) the Dirac function. Therefore, the Ito lemma at t = T would provide d(ST − K)+ = H(ST − K)dST +

1 v S2 �(S − K)dT. 2 T T T

Using the fact that the forward price (here with zero interest rates, the stock price) is a martingale under the risk-neutral measure dV = dE[(ST − K)+ ] =

1 E[vT S2T �(ST − K)]dT, 2

now we have E[vT S2T �(ST − K)] = E[vT |ST = K]K2 E[�(ST − K)] = E[vT |ST = K]K2

�2V . �K2

Putting all this together �V 1 �2V = K2 2 E[vT |ST = K] �T 2 �K and by the above expression of � 2 (K, T), we will have � 2 (K, T) = E[vT |ST = K] as claimed.

(QED)

The Derman Kani Approach The Derman Kani technique is very similar to the Dupire approach, except it uses the Binomial (or Trinomial) Tree framework instead of the continuous one.

18

INSIDE VOLATILITY FILTERING

Using the Binomial Tree notations, their upward transition probability pi from the spot si at time tn to the upper node Si+1 at the following time-step tn+1 is obtained from the usual pi =

Fi − Si Si+1 − Si

(1.23)

where Fi is the stock forward price known from the market, and Si the lower spot at the step tn+1 . In addition, we have for a call expiring at time-step tn+1 C(K, tn+1 ) = e−rΔt

n ∑ [�j pj + �j+1 (1 − pj+1 )]MAX(Sj+1 − K, 0) j=1

where �j ’s are the known Arrow-Debreu prices corresponding to the discounted probability of getting to the point sj at time tn from S0 , the initial stock price. These probabilities could easily be derived iteratively. This allows us after some calculation to obtain Si+1 as a function of si and Si , namely Si+1 =

Si [erΔt C(si , K, tn+1 ) − Σ] − �i si (Fi − Si ) [erΔt C(si , K, tn+1 ) − Σ] − �i (Fi − Si )

∑ where the term Σ represents the sum nj=i+1 �j (Fj − si ). This means that after choosing the usual centering condition for the Binomial Tree s2i = Si Si+1 , we have all the elements to build the tree and deduce the implied distribution from the Arrow-Debreu prices.

Stability Issues The local volatility models are very elegant and theoretically sound; however, they present in practice many stability issues. They are Ill-Posed Inversion problems and are extremely sensitive to the input data.15 This might introduce arbitrage opportunities and in some cases negative probabilities or variances. Derman and Kani suggest overwriting techniques to avoid such problems. 15

See Tavella [237] or Avellaneda [17].

19

The Volatility Problem

Andersen [14] tries to improve this issue by using an Implicit Finite Difference method. However, he recognizes that the negative variance problem could still happen. One way to make the results smoother is to use a constrained optimization. In other words, when trying to it theoretical results Ctheo to the market prices Cmrkt , instead of minimizing N ∑

(Ctheo (Kj ) − Cmrkt (Kj ))2

j=1

we could minimize N

�

�� ∑ (C (K ) − Cmrkt (Kj ))2 + �t j=1 theo j

where � is a constraint parameter, which could also be interpreted as a Lagrange multiplier. However, this is an artiicial way to smoothen the results and the real issue remains that once again, we have an inversion problem that is inherently unstable. What is more, local volatility models imply that future implied volatility smiles will be lat relative to today’s, which is another limitation.16 As we will see in the following section, stochastic volatility models offer more time-homogeneous volatility smiles. An alternative approach suggested in [17] would be to choose a prior risk-neutral distribution for the asset (based on a subjective view) and then minimize the relative Entropy distance between the desired surface and this prior distribution. This approach uses the Kullback-Leibler distance (which we will discuss in the context of MLE) and performs the minimization via Dynamic Programming [37] on a tree.

Calibration Frequency One of the most attractive features of local-vol models is their ability to match plain-vanilla puts and calls exactly. This will avoid arbitrage situations, or worse, market manipulations by traders to create “phantom” proits.

16

See Gatheral [119].

20

INSIDE VOLATILITY FILTERING

As explained in Hull [154], these arbitrage-free models were developed by researchers with a Single Calibration (SC) methodology assumption. However, traders use them with a Continual Recalibration (CR) strategy in practice. Indeed, if they used the SC version of the model, signiicant errors would be introduced from one week to the following, as shown by Dumas et al. [93]. However, once this CR version is used, there is no guarantee that the no-arbitrage property of the original SC model is preserved. Indeed, the Dupire equation determines the marginal stock distribution at different points in time, but not the joint distribution of these stock prices. Therefore, a path-dependent option could very well be mispriced, and the more path-dependent this option, the greater the mispricing. Hull [154] takes the example of a Bet Option, a Compound Option, and a Barrier Option. The Bet Option depends on the distribution of the stock at one point in time and therefore is correctly priced with a Continually Recalibrated Local-Vol model. The Compound Option has some path dependency, hence a certain amount of mispricing compared to a stochastic volatility (SV) model. Finally, the Barrier Option has a strong degree of path dependency and will introduce large errors. Note that this is due to the discrete nature of the data. Indeed, the maturities we have are limited. If we had all possible maturities in a continuous way, the joint distribution would be determined completely. Also, when interpolating in time, it is customary to interpolate upon the true variance t�t2 rather than the volatility �t given the equation T2 � 2 (T2 ) = T1 � 2 (T1 ) + (T2 − T1 )� 2 (T1 , T2 ). Interpolating upon the true variance will provide smoother results as shown by Jackel [159]. Proof

Indeed, calling for 0 ≤ T1 ≤ T2 the spot return variances Var(0, T2 ) = T2 � 2 (T2 ) Var(0, T1 ) = T1 � 2 (T1 )

for a Brownian Motion, we have independent increments and therefore a forward variance Var(T1 , T2 ) such that Var(0, T1 ) + Var(T1 , T2 ) = Var(0, T2 ) which demonstrates the point. (QED)

21

The Volatility Problem

STOCHASTIC VOLATILITY Unlike non-parametric local volatility models, parametric stochastic volatility (SV) models deine a speciic stochastic differential equation for the unobservable instantaneous variance. As we shall see, the previously deined CEV model could be considered a special case of these models.

Stochastic Volatility Processes The idea would be to use a different stochastic process for � altogether. Making the volatility a deterministic function of the spot is a special “degenerate” two-factor, a natural generalization of which would precisely be to have two stochastic processes with a non-perfect correlation.17 Several different stochastic processes have been suggested for the volatility. One popular one is the Ornstein-Uhlenbeck (OU) process: d�t = −��t dt + �dZt

(1.24)

where � and � are two parameters; remembering the stock equation dSt = �t St dt + �t St dBt , there is a (usually negative) correlation � between dZt and dBt which can in turn be time or level dependent. Heston [141] and Stein [234] were among those who suggested the use of this process. Using Ito’s lemma, we can see that the stock-return variance vt = �t2 satisies a Square-Root or Cox-Ingersoll-Ross (CIR) process √ dvt = (� − �vt )dt + � vt dZt (1.25) with � = � 2 , � = 2� and � = 2�. Note that the OU process has a closed-form solution �t = �0 e−�t + �

∫0

t

e−�(t−s) dZs 2

� (1 − e−2�t )), with Φ again the which means that �t follows in law Φ(�0 e−�t , 2� normal distribution. This was discussed in Fouque [109] and Shreve [229].

17

Note that here, the instantaneous volatility is stochastic. Recent work by researchers such as Schonbucher supposes a Stochastic Implied-Volatility process, which is a completely different approach. See for instance [224]. On the other hand, Avellaneda et al. [18] use the concept of uncertain volatility for pricing and hedging derivative securities. They make the volatility switch between two extreme values based on the convexity of the derivative contract and obtain a nonlinear Black-Scholes-Barenblatt equation, which they solve on a grid.

22

INSIDE VOLATILITY FILTERING

Heston and Nandi [144] show that this process corresponds to a special case of the General Auto-Regressive Conditional Heteroskedasticity (GARCH) model that we will discuss further in the next section. Another popular process is the GARCH(1,1) process, where we would have dvt = (� − �vt )dt + �vt dZt . (1.26)

GARCH and Diffusion Limits The most elementary GARCH process called GARCH(1,1) was developed originally in the ield of econometrics by Engle [99] and Bollerslev [42] in a discrete framework. The stock discrete equation (1.3) could be rewritten by taking Δt = 1 and vn = �n2 as ( ) √ 1 (1.27) ln Sn+1 = ln Sn + � − vn+1 + vn+1 Bn+1 2 and calling the mean adjusted return ) ( ) ( √ Sn 1 − � − vn = vn Bn . un = ln Sn−1 2

(1.28)

The variance process in GARCH(1,1) is supposed to be vn+1 = �0 + �vn + �u2n = �0 + �vn + �vn B2n

(1.29)

where � and � are weight parameters and �0 a parameter related to the long-term variance.18 Nelson [204] shows that as the time interval length decreases and becomes ininitesimal, equation (1.29) becomes precisely the previously cited equation (1.26). To be more accurate, there is a weak convergence of the discrete GARCH process to the continuous diffusion limit.19 For a GARCH(1,1) continuous diffusion, the correlation between dZt and dBt is zero. 18 It is worth mentioning that as explained in [105] a GARCH(1,1) model could be rewritten as an Auto-Regressive Moving Average model of irst order ARMA(1,1) and therefore an Auto-Regressive model of ininite order AR(+∞). GARCH is therefore a parsimonious model that can it the data with only a few parameters. Fitting the same data with an ARCH or AR model would require a much larger number of parameters. This feature makes the GARCH model very attractive. 19 For an explanation on weak convergence see for example Varadhan [241].

23

The Volatility Problem

It might appear surprising that even if the GARCH(1,1) process has only one source of randomness, namely Bn , the continuous version has two independent Brownian Motions. This is understandable if we consider Bn a standard normal random variable and An = B2n − 1 another random variable. It is fairly easy to see that An and Bn are uncorrelated even if An is a function of Bn . As we go toward the continuous version, we can use Donsker’s theorem,20 by letting the time interval Δt → 0, to prove that we end up with two uncorrelated and therefore independent Brownian Motions. This is a limitation of the GARCH(1,1) model, hence the introduction of the Nonlinear Asymmetric GARCH (NGARCH) model. Duan [88] attempts to explain the volatility smile by using the NGARCH process expressed by √ vn+1 = �0 + �vn + �(un − c vn )2

(1.30)

where c is a parameter to be determined. The NGARCH process was irst introduced by Engle [102]. The continuous counterpart of NGARCH is the same equation (1.26), except unlike the equation resulting from GARCH(1,1) there is a non-zero correlation between the stock process and the volatility process. This correlation is created precisely because of the parameter c that was introduced, and is once again called the leverage effect. The parameter c is sometimes referred to as the leverage parameter. We can ind the following relationships between the discrete process and the continuous diffusion limit parameters by letting the time interval become ininitesimal �=

�0 dt2

1 − �(1 + c2 ) − � dt √ � − 1 + 4c2 �=� dt

�=

and the correlation between dBt and dZt −2c �= √ � − 1 + 4c2

20

For a discussion on Donsker’s theorem, similar to the central limit theorem, see for instance Whitt [247].

24

INSIDE VOLATILITY FILTERING

where � represents the Pearson kurtosis21 of the mean adjusted returns (un ). As we can see, the sign of the correlation � is determined by the parameter c. Proof A quick proof of the convergence to diffusion limit could be outlined as follows. Let us assume that c = 0 for simplicity, we therefore are dealing with the GARCH(1,1) case. As we saw vn+1 = �0 + �vn + �vn B2n therefore or

vn+1 − vn = �0 + �vn − vn + �vn − �vn + �vn B2n vn+1 − vn = �0 − (1 − � − �)vn + �vn (B2n − 1).

Now,√allowing the time-step Δt to become variable and posing Zn = (B2n − 1)∕ � − 1 √ vn+Δt − vn = �Δt2 − �Δtvn + �vn ΔtZn and annualizing vn by posing vt = vn ∕Δt we shall have √ vt+Δt − vt = �Δt − �Δtvt + �vt ΔtZn and as Δt → 0 we get dvt = (� − �vt )dt + �vt dZt as claimed.

(QED)

Note that the discrete GARCH version of the Square-Root process (1.15) is √ (1.31) vn+1 = �0 + �vn + �(Bn − c vn )2 as Heston and Nandi show22 in [144]. Also, note that having a diffusion process dvt = b(vt )dt + a(vt )dZt we can apply an Euler approximation23 to discretize √ and obtain a Monte-Carlo process such as vn+1 − vn = b(vn )Δt + a(vn ) ΔtZn . It is important to note that if we use a GARCH process and go to the continuous diffusion limit, and 21 The kurtosis corresponds to the fourth moment. The Pearson kurtosis for a normal distribution is equal to 3. 22 For a detailed discussion on the convergence of different GARCH models toward their diffusion limits, also see Duan [90]. 23 See for instance Jones [173].

25

The Volatility Problem

then apply an Euler Approximation, we will not necessarily ind the original GARCH process again. Indeed, there are many different ways to discretize the continuous diffusion limit and the GARCH process corresponds to one special way. In particular, if we use (1.31) and allow Δt → 0 to get to the continuous diffusion limit, we shall obtain (1.25). As we will see later in the section on Mixing Solutions, we can then apply a discretization to this process and obtain a Monte-Carlo simulation √ √ vn+1 = vn + (� − �vn )Δt + � vn ΔtZn which is again different from (1.31) but obviously has to be consistent in terms of pricing. However, we should know which method we are working with from the very beginning to perform our calibration on the corresponding speciic process. Corradi [66] explains this in the following manner: The discrete GARCH model could converge either toward a two-factor continuous limit if one chooses the Nelson parameterization, or could very well converge to a one-factor diffusion limit if one chooses another parameterization. What is more, an appropriate Euler discretization of the one-factor continuous model will provide a GARCH discrete process, while as previously mentioned the discretization of the two-factor diffusion model provides a two-factor discrete process. This distinction is fundamental and could explain why GARCH and SV behave differently in terms of parameter estimation. 160

Square-Root Model via GARCH Market Model

140

Call Price

120 100 80 60 40 20 0 950

1000

1050 Strike Price

1100

1150

GARCH Monte-Carlo simulation with the Square-Root model for SPX on February 12, 2002, with Index = 1107.5 USD, one month to maturity. Powell Optimization method was used for Least-Square Calibration.

26

INSIDE VOLATILITY FILTERING

THE PRICING PDE UNDER STOCHASTIC VOLATILITY A very important issue to underline here is that, because of the unhedgeable second source of randomness, the concept of Market Completeness is lost. We can no longer have a straightforward risk neutral pricing. This is where the market price of risk will come into consideration.

The Market Price of Volatility Risk Indeed, taking a more general form for the variance process dvt = b(vt )dt + a(vt )dZt

(1.32)

as we previously said, using the Black-Scholes risk-neutrality argument, the equation (1.1) could be replaced with dSt = (rt − qt )St dt + �t St dBt .

(1.33)

This is equivalent to changing the probability measure from the real one to the risk-neutral one.24 We therefore need to use (1.33) together with the risk adjusted volatility process ̃ )dt + a(v )dZ dvt = b(v t t t where

(1.34)

̃ ) = b(v ) − �a(v ) b(v t t t

with � the market price of volatility risk. This quantity is closely related to the market price of risk for the stock �e = (� − r)∕�. Indeed, as Hobson [147] and Lewis [185] both show, we have � = ��e +

√

1 − � 2 �∗

(1.35)

where �∗ is the market price of risk associated with dBt − �dZt , which can also be regarded as the market price of risk for the hedged portfolio. The passage from equation (1.32) to (1.34) and the introduction of the market price of volatility risk could also be explained by the Girsanov theorem as was done for instance in Fouque [109]. 24

See Hull [153] or Shreve [229] for more detail.

27

The Volatility Problem

It is important to underline the difference between the real and the risk-neutral measures here. If we use historic stock prices together with the real stock-return drift � to estimate the process parameters, we will obtain the real volatility drift b(v). An alternative method would be to ̃ estimate b(v) by using current option prices and performing a least square estimation. These calibration methods will be discussed in more detail in the following chapters. The risk-neutral version for a discrete NGARCH model would also involve the market price of risk and instead of the usual ) ( √ 1 ln Sn+1 = ln Sn + � − vn+1 + vn+1 Bn+1 2 vn+1 = �0 + �vn + �vn (Bn − c)2 , we would have (

ln Sn+1

1 = ln Sn + r − vn+1 2

) +

√ vn+1 B̃ n+1

(1.36)

vn+1 = �0 + �vn + �vn (B̃ n − c − �e )2 where B̃ n = Bn + �e which could be regarded as the discrete version of the Girsanov theorem. Note that the market price of risk for the stock �e is not separable from the leverage parameter c in this formulation. Duan shows in [89] and [91] that the risk-neutral GARCH system (1.36) will indeed converge toward the continuous risk-neutral GARCH √ dSt = St rdt + St vt dBt ̃ t )dt + �vt dZt dvt = (� − �v as we expected.

The Two-Factor PDE From here, writing a two-factor PDE for a derivative security f becomes a simple application of the two-dimensional Ito’s lemma. The PDE will be25 rf =

25

2 �2 f �f �f 1 ̃ �f + 1 a2 (v) � f + (r − q)S + vS2 2 + b(v) �t �S 2 �S �v 2 �v2 √ �2f + �a(v) vS �S�v

For a proof of the derivation see Wilmott [249] or Lewis [185].

(1.37)

28

INSIDE VOLATILITY FILTERING

Therefore, it is possible, after calibration, to apply a Finite Difference method26 to the above PDE to price the derivative f (S, t, v). An alternative would be to use directly the stochastic processes for dSt and dvt and apply a two-factor Monte-Carlo simulation. Later in the chapter we will also mention other possible methods such as the Mixing Solution or Asymptotic Approximations. Other possible approaches for incomplete markets and stochastic volatility assumption include Super-Replication and Local Risk Minimization.27 The Super-Replication strategy is the cheapest self-inancing strategy with a terminal value no less than the payoff of the derivative contract. This technique was primarily developed by El-Karoui and Quenez in [96]. Local Risk Minimization involves a partial hedging of the risk. The risk is reduced to an “intrinsic component” by taking an offsetting position in the underlying security as usual. This method was developed by Follmer and Sondermann in [107].

THE GENERALIZED FOURIER TRANSFORM The Transform Technique One useful technique to apply to the PDE (1.37) is the Generalized Fourier Transform.28 First, we can use the variable x = ln S in which case, using Ito’s lemma, (1.37) could be rewritten as ( ) 2 �f �f 1 1 �2f ̃ �f + 1 a2 (v) � f rf = + r−q− v + v 2 + b(v) �t 2 �x 2 �x �v 2 �v2 √ �2f + �a(v) v �x�v Calling f̂(k, v, t) =

(1.38)

+∞

∫−∞

eikx f (x, v, t)dx

(1.39)

where k is a complex number,29 f̂ will be deined in a complex strip where the imaginary part of k is between two real numbers � and �. 26 See for instance Tavella [238] or Wilmott [249] for a discussion on Finite Difference Methods. 27 For a discussion on both these techniques see Frey [112]. 28 See Lewis [185] for a√ detailed discussion on this technique. 29 As usual we note i = −1.

29

The Volatility Problem

Once f̂ is suitably deined, meaning that ki = (k) (the imaginary part of k) is within the appropriate strip, we can write the Inverse Fourier Transform iki +∞ 1 f (x, v, t) = e−ikx f̂(k, v, t)dk (1.40) 2� ∫iki −∞ where we are integrating for a ixed ki parallel to the real axis. Each derivative satisfying (1.37) or equivalently (1.38) has a known payoff G(ST ) at maturity. For instance, as we said before, a call option has a payoff MAX(0, ST − K) where K is the call strike price. It is easy to see that for ki > 1 the Fourier Transform of a call option exists and the payoff transform is Kik+1 − 2 (1.41) k − ik Proof Indeed, we can write +∞

∫−∞

as stated.

+∞

eikx (ex − K)dx ∫ln K ( ik+1 ) Kik K =0− −K ik + 1 ik ( ) 1 1 1 − = −Kik+1 2 = −Kik+1 ik + 1 ik k − ik

eikx (ex − K)+ dx =

(QED)

The same could be applied to a put option or other derivative securities. In particular, a covered call (stock minus call) having a payoff MIN(ST , K) will have a transform for 0 < ki < 1 equal to Kik+1 k2 − ik

(1.42)

Applying the transform to the PDE (1.38) and introducing � = T − t and ̂ v, �) = e(r+ik(r−q))� f̂(k, v, �) h(k,

(1.43)

and posing30 c(k) = 12 (k2 − ik), we get the new PDE equation √ � ĥ � 2 ĥ 1 � ĥ ̃ ̂ = a2 (v) 2 + (b(v) − ik�(v)a(v) v) − c(k)vh. �� 2 �v �v 30

We are following Lewis [185] notations.

(1.44)

30

INSIDE VOLATILITY FILTERING

̂ v, �) satisfying Lewis calls the Fundamental Transform a function H(k, ̂ the PDE (1.44) and satisfying the initial condition H(k, v, � = 0) = 1. If we know this Fundamental Transform, we can then multiply it by the derivative security’s payoff transform and then divide it by e(r+ik(r−q))� and apply the inverse Fourier technique by keeping ki in an appropriate strip and inally get the derivative as a function of x = ln S.

Special Cases There are cases where the Fundamental Transform is known. The case of a constant (or deterministic) volatility is the most elementary one. Indeed, using (1.44) together with dvt = 0 we can easily ind ̂ v, �) = e−c(k)v� H(k, which is analytic in k over the entire complex plane. Using the call payoff transform (1.41), we can rederive the Black-Scholes equation. The same can be done if we have a deterministic volatility dvt = b(vt )dt by using the function Y(v, t) where dY = b(Y)dt. The Square-Root model (1.25) is another important case where ̂ v, �) is known and analytic. We have for this process H(k, √ dvt = (� − �vt )dt + � vt dZt or under the risk-neutral measure √ ̃ t )dt + � vt dZt dvt = (� − �v √ with �̃ = (1 − �)�� + � 2 − �(1 − �)� 2 where � ≤ 1 represents the riskaversion factor. For the Fundamental Transform, we get ̂ v, �) = exp[f1 (t) + f2 (t)v] H(k, with t=

2 2 � c̃ = 2 c(k) and �2 � [ ( )] 1 − hetd f1 (t) = tg − ln �̃ 1−h [ ] 1 − etd g f2 (t) = 1 − hetd

1 2 � � 2

�̃ =

(1.45)

31

The Volatility Problem

where √

1 �+d (� + d) h = 2 �−d √ 2 � = 2 [(1 − � + ik)�� + � 2 − �(1 − �)� 2 ] �

d=

2

� + 4̃c

g=

and

This transform has a cumbersome expression, but it can be seen that it is analytic in k and therefore always exists. For a proof refer to Lewis [185]. The Inversion of the Fourier Transform for the Square-Root (Heston) model is a very popular and powerful approach. It is appealing due to its robustness and speed. The following example is based on SPX options as of March 9, 2004, expiring in one to eight years from the calibration date. SPX implied surface as of 03/09/2004. T is the maturity and M = K∕S the inverse of the moneyness. T / M 0.70

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.30

1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000

21.29 18.73 18.69 18.48 18.12 17.90 17.86 17.54

19.73 18.68 17.96 17.87 17.70 17.56 17.59 17.37

18.21 17.65 17.28 17.33 17.29 17.23 17.30 17.17

16.81 16.69 16.61 16.78 16.88 16.90 17.00 16.95

15.51 15.79 15.97 16.26 16.50 16.57 16.71 16.72

14.43 14.98 15.39 15.78 16.13 16.25 16.43 16.50

13.61 14.26 14.86 15.33 15.77 15.94 16.15 16.27

13.12 13.67 14.38 14.92 15.42 15.64 15.88 16.04

12.94 13.22 13.96 14.53 15.11 15.35 15.62 15.82

13.23 12.75 13.30 13.93 14.54 14.83 15.15 15.40

1.15

24.61 21.94 20.16 19.64 18.89 18.46 18.32 17.73

Heston prices itted to the March 9, 2004, surface. T / M 0.70

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000

21.44 22.98 24.18 25.48 26.54 27.55 28.61 29.34

17.09 18.98 20.44 21.93 23.20 24.34 25.52 26.39

13.01 15.25 16.98 18.66 20.09 21.36 22.64 23.64

9.33 11.87 13.82 15.63 17.22 18.60 19.96 21.07

6.18 8.89 11.00 12.91 14.63 16.08 17.49 18.69

3.72 6.37 8.55 10.50 12.30 13.79 15.24 16.51

2.03 1.03 0.50 4.35 2.83 1.78 6.47 4.77 3.43 8.39 6.61 5.10 10.21 8.39 6.82 11.73 9.89 8.26 13.19 11.35 9.70 14.51 12.68 11.04

30.67 31.60 32.31 33.21 33.87 34.56 35.35 35.77

1.20 1.30 0.13 0.66 1.66 2.93 4.36 5.64 6.97 8.24

32

INSIDE VOLATILITY FILTERING

As we shall see further, the optimal Heston parameter-set to it this surface could be found via a Least Square Estimation approach, and for the index at S = 1156.86 USD we ind the optimal parameters v̂ 0 = 0.1940 and ̂ = (�, ̂ �, ̂ �) Ψ ̂ �, ̂ = (0.052042332, 1.8408, 0.4710, −0.4677). 25.00 20.00 15.00 j 10.00 5.00

115%

130%

K/S

105%

95%

85%

70%

0.00

SPX implied surface as of March 9, 2004. We can observe the negative skewness as well as the lattening of the slope with maturity.

THE MIXING SOLUTION The Romano Touzi Approach The idea of Mixing Solutions was probably presented for the irst time by Hull and White [156] for a zero correlation case. Later, Romano and Touzi [220] generalized this approach for a correlated case. The basic idea is to separate the random processes of the stock and the volatility, integrate the stock process conditionally on a given volatility, and inally end up with a one-factor problem. Let us recall the two processes we had: dSt = (rt − qt )St dt + �t St dBt and

̃ )dt + a(v )dZ dvt = b(v t t t

under a risk-neutral measure.

33

The Volatility Problem

Given a correlation �t between dBt and dZt , we can introduce the Brownian Motion dWt independent of dZt and write the usual Cholesky31 factorization: √ dBt = �t dZt + 1 − �2t dWt We can then introduce the same Xt = ln St and write the new system of equations: √ 1 (1.46) dXt = (r − q)dt + dYt − (1 − �2t )�t2 dt + 1 − �2t �t dWt 2 1 dYt = − �2t �t2 dt + �t �t dZt 2 dv = b̃ dt + a dZ t

t

t

t

where once again, the two Brownian Motions are independent. It is now possible to integrate the stock process for a given volatility and end up with an expectation on the volatility process only. We can think of (1.46) as the limit of a discrete process, while the time step Δt → 0. For a derivative security f (S0 , v0 , T) with a payoff32 G(ST ), using the bivariate normal density for two uncorrelated variables, we can write f (S0 , v0 , T) = e−rT E0 [G(ST )] ∞

Δt→0 ∫−∞

= e−rT lim T−Δt

×

∏ t=0

(1.47) ∞

...

∫−∞

G(ST )

] [ dZt dWt 1 2 2 . exp − (Zt + Wt ) 2 2�

Now if we know how to integrate this over dWt for a given volatility and we know the result f ∗ (S, v, T) (for instance, for a European call option, we know the Black-Scholes formula (1.6), there are many other cases where we have closed-form solutions), then we can introduce the auxiliary variables33 ) ( T T 1 (1.48) � � dZ �2 � 2 dt + Seff = S0 eYT = S0 exp − ∫0 t t t 2 ∫0 t t 31

See for example Press [214]. The payoff should not depend on the volatility process. 33 Again, all notations are taken from Lewis [185].

32

34

INSIDE VOLATILITY FILTERING

and veff =

T

1 (1 − �2t )�t2 dt T ∫0

(1.49)

and as Romano and Touzi prove in [220] we will have f (S0 , v0 , T) = E0 [f ∗ (Seff , veff , T)]

(1.50)

where this last expectation is being taken on dZt only. Note that in the zero correlation case discussed by Hull and White [156] T we have Seff = S0 and veff = vT = T1 ∫0 �t2 dt, which makes the expression (1.50) a natural weighted average.

A One-Factor Monte-Carlo Technique As Lewis suggests, this will enable us to run a single-factor Monte-Carlo simulation on the dZt and apply the known closed-form for each simulated path. The method does suppose however that the payoff G(ST ) does not depend on the volatility. Indeed, going back to (1.46) we can do a simulation on Yt and vt using the random sequence of (Zt ), then after one path is generated, we can cal∑T−Δt (1 − �2t )vt Δt and then apply the culate Seff = S0 exp(YT ) and veff = T1 t=0 known closed-form (e.g., Black-Scholes for a call or put) with Seff and veff . Repeating this procedure for a large number of times and averaging over the paths, as we usually do in Monte-Carlo methods, we will have f (S0 , v0 , T). This will give us a way to calibrate the model parameters to the market data. For instance using the Square-Root model √ dvt = (� − �vt )dt + � vt dZt , we can estimate �, �, �, and � from the market prices via a least-square estimation applied to theoretical prices obtained from the above Monte-Carlo method. We can either use a single calibration and suppose we have time-independent parameters, or perform one calibration per maturity. The single calibration method is known to provide a bad it, hence the idea of adding jumps to the stochastic volatility process as described by Matytsin [197]. However, this method will introduce new parameters for calibration.34 34

Eraker et al. [103] claim that a model containing jumps in the return and the volatility process will it the options and the underlying data well, and will have no misspeciication left.

35

The Volatility Problem Volatility Smile 200 180

Market 1 Month to Maturity Model Market 7 Months to Maturity Model

160

Call Price

140 120 100 80 60 40 20 0 950

1000

1050 Strike Price

1100

1150

Mixing Monte-Carlo simulation with the Square-Root model for SPX on February 12, 2002, with Index = 1107.5 USD, one month and seven months to maturity. Powell Optimization method was used for Least-Square Calibration. As we can see, both maturities are itted fairly well.

THE LONG-TERM ASYMPTOTIC CASE In this section, we will discuss the case where the contract time to maturity is very large, that is, t → ∞. We will focus on the special case of a Square-Root process, since this is the model we will use in many cases.

The Deterministic Case We shall start with the case of deterministic volatility and use that for the more general case of the stochastic volatility. We know that under the Square-Root model the variance follows √ dvt = (� − �vt )dt + � vt dZt . As an approximation, we can drop the stochastic term and obtain dvt = � − �vt dt which is an ordinary differential equation providing us immediately with ) ( � −�t � e (1.51) vt = + v − � � where v is the initial variance for t = 0.

36

INSIDE VOLATILITY FILTERING

Using the results from the Fundamental Transform for a covered call option and put-call parity, we have for 0 < ki < 1 call(S, v, �) = Se−q� − Ke−r�

iki +∞ ̂ v, �) 1 H(k, e−ikX 2 dk 2� ∫iki −∞ k − ik

(1.52)

−q�

Se where � = T − t, and X = ln( Ke −r� ) represents the adjusted moneyness of the option. For the special “At the Money”35 case where X = 0 we have [ ] iki +∞ ̂ 1 H(k, v, �) call(S, v, �) = Ke−r� 1 − dk . (1.53) 2� ∫iki −∞ k2 − ik

As we previously said for a deterministic volatility case, we know the Fundamental Transform ̂ v, �) = exp[−c(k)U(v, �)] H(k, with U(v, �) = ∫0 v(t)dt and as before c(k) = 12 (k2 − ik), which in the special case of the Square-Root model (1.51), will provide us with ) )( ( 1 − e−�� � � . U(v, �) = � + v − � � � �

̂ This shows once again that H(k) is analytic in k over the entire complex plane. Now if we let � → ∞ we can write the approximation [ iki +∞ ( )] � 1 1 � dk call(S, v, �) v− ≈1− exp −c(k) � − c(k) . −r� 2 ∫ Ke 2� iki −∞ � � � k − ik (1.54) We can either calculate this integral exactly using the BlackScholes theory, or take the minimum where c′ (k0 ) = 0, meaning k0 = 2i , and perform a Taylor approximation parallel to the real axis around the point k = kr + 2i , which will give us [ ( )] ( ) � 2 � 1 call(S, v, �) v− ≈ 1 − exp − � exp − Ke−r� � 8� 8� � ∞ ( ) � × exp −k2r � dkr ∫−∞ 2� 35

This is different from the usual deinition of At the Money calls where S = K. This vocabulary is borrowed from Alan Lewis.

37

The Volatility Problem

The integral being a Gaussian, we will get the result call(S, v, �) ≈1− Ke−r�

√

[ ( )] ) ( � 1 � 8� v− exp − exp − � ��� 8� � 8�

(1.55)

which inishes our deterministic approximation case.

The Stochastic Case For the stochastic volatility (SV) case, Lewis uses the same Taylor expansion. He notices that for the deterministic case we had ̂ v, �) = exp[−c(k)U(v, �)] ≈ exp[−�(k)�]u(k, v) H(k, for � → ∞ where �(k) = c(k) and

� �

[ ( )] � 1 v− u(k, v) = exp −c(k) . � �

If we suppose that this identity holds for the SV case as well, we can use the PDE (1.44) and interpret the result as an eigenvalue-eigenfunction identity with the eigenvalue �(k) and the eigenfunction u(k, v). This assumption is reasonable since the irst Taylor approximation term for the stochastic process is deterministic. Indeed, introducing the operator √ du d2 u 1 ̃ − ik�(v)a(v) v] + c(k)vu Λ(u) = − a2 (v) 2 − [b(v) 2 dv dv we have (1.56)

Λ(u) = �(k)u.

Now the idea would be to perform a Taylor expansion around the minimum k0 where �′ (k0 ) = 0. Lewis shows that such k0 is always situated on the imaginary axis. This property is referred to as the “ridge” property. The Taylor expansion along the real axis could be written as �(k) = �(k0 + kr ) ≈ �(k0 ) +

1 2 ′′ k � (k0 ). 2 r

38

INSIDE VOLATILITY FILTERING

Note that we are dealing with a minimum and therefore �′′ (k0 ) > 0. Using the previous second-order approximation for �(k) we get u(k , v) call(S, v, �) 1 ≈1− 2 0 exp[−�(k0 )�]. √ Ke−r� k0 − ik0 2��′′ (k0 )� We can then move from the special “At the Money” case to the general Se−q� case by reintroducing X = ln( Ke −r� ) and we will inally obtain u(k , v) call(S, v, �) 1 ≈ eX − 2 0 exp[−�(k0 )� − ik0 X] √ Ke−r� k0 − ik0 2��′′ (k0 )�

(1.57)

which completes our determination of the asymptotic closed-form in the general case. For the special case of the Square-Root model, taking the risk-neutral case � = 1, we have36 �(k) = −�g∗ (k) =

� √ [ (� + ik��)2 + (k2 − ik)� 2 − (� + ik��)] �2

which also allows us to calculate �′′ (k). Also u(k, v) = exp[g∗ (k)v] where we use the notations from (1.45) and we pose g∗ = g − d. The k0 such that �′ (k0 ) = 0 is k0 =

i 1 − �2

(

]) [ 1√ 2 1 � 4� + � 2 − 4��� − �− 2 � 2

which together with (1.57) provides us with the result for call(S, v, �) in the asymptotic case under the Square-Root stochastic volatility model. Note that for � → 0 and � → 0 we ind again the deterministic result k0 → 2i . √ We can go back to the general case � ≤ 1 by replacing � with � 2 − �(1 − �)� 2 + (1 − �)�� since this transformation is independent from k altogether. 36

39

The Volatility Problem

A Series Expansion on Volatility-of-Volatility Another asymptotic approach for the stochastic volatility model suggested by Lewis [185] is a Taylor expansion on the volatility-of-volatility. There are two possibilities for this; we can perform the expansion either for the option price or for the implied-volatility directly. In what follows, we consider the former approach. Once again, we use the fundamental transform H(k, V, �) with H(k, V, 0) = 1 and √ �H �2H 1 �H ̃ = a2 (v) 2 + (b(v) − ik�(v)a(v) v) − c(k)vH �� 2 �v �v and c(k) = 21 (k2 − ik). We then pose a(v) = ��(v) and expand H(k, V, �) on powers of � and inally apply the inverse Fourier Transform to obtain an expansion on the call price. ( ) With our usual notations � = T − t, X = ln KS + (r − q)� and Z(V) = V�, the series will be ̃ 11 �cBS (S, v, �) C(S, V, �) = cBS (S, v, �) + �� −1 J1 R �V ] [ ̃ 22 �cBS (S, v, �) + O(� 3 ) ̃ 20 + � −1 J4 R ̃ 12 + 1 � −2 J2 R + � 2 � −2 J3 R 1 2 �V where v(V, �) is the deterministic variance v(V, �) =

) )( ( � 1 − e−�� � + V− , � � ��

̃ pq = Rpq (X, v(V, �), �) with Rpq given polynomials of (X, Z) of degree and R four at most, and Jn ’s known functions of (V, �). The explicit expressions for all these functions are given in the third chapter of the Lewis book [185]. The obvious advantage of this approach is its speed and stability. The issue of lack of time-homogeneity of the parameters Ψ = (�, �, �, �) could be addressed by performing one calibration per time-interval. In this case, for each time-interval [tn , tn+1 ] we will have one set of parameters Ψn = (�n , �n , �n , �n ) and depending on what maturity T we are dealing with, we will use one or the other parameter-set. We compare the values obtained from this series-based approach with those from a mixing Monte-Carlo method in Figure 1.1. We are taking the example that Heston studied in [141]. The graph shows the difference

40

INSIDE VOLATILITY FILTERING Heston Prices via Mixing and Vol-of-Vol Series 0.15

Price Difference

0.1 0.05 0 ‒0.05 ‒0.1 ‒0.15 70

Mixing Vol-of-Vol Series 80

90

100

110

120

130

Stock (USD)

FIGURE 1.1 Comparing the volatility-of-volatility series expansion with the Monte-Carlo mixing model. The graph shows the price difference C(S, V, �) − cBS (S, V, �). We are taking � = 0.10 and � = −0.50. This example was used in the original Heston paper.

C(S, V, �) − cBS (S, V, �) for a ixed K = 100 USD and � = 0.50 years. The other inputs are � = 0.02, � = 2.00, � = 0.10, � = −0.50, V = 0.01, and r = q = 0. As we can see, the true value of the call is lower than the Black-Scholes value for the OTM region. The higher � and |�| are, the larger this difference will be. In Figures 1.2 and 1.3 we reset the correlation � to zero to have a symmetric distribution, but we use a volatility-of-volatility of � = 0.10 and � = 0.20 respectively. As discussed, the parameter � is the one creating the leptokurticity phenomenon. A higher volatility-of-volatility causes higher valuation for Far-from-the-Money options.37 Unfortunately, the series approximation in Figure 1.1 becomes poor as soon as the volatility of volatility becomes larger than 0.40 and the maturity becomes of the order of one year. This case is not unusual at all and therefore makes the use of this method limited. This is why the method of choice remains the Inversion of the Fourier Transform, as previously described.

37 Also note that the gap between the closed-form series and the Monte-Carlo model increases with �. Indeed the accuracy of the expansion decreases as � becomes larger.

41

The Volatility Problem

Heston Prices via Mixing and Vol-of-Vol Series 0.015 0.01

Price Difference

0.005 0 ‒0.005 ‒0.01 ‒0.015 ‒0.02 Mixing Vol-of-Vol Series

‒0.025 ‒0.03 70

80

90

100

110

120

130

Stock (USD)

FIGURE 1.2 Comparing the volatility-of-volatility series expansion with the Monte-Carlo mixing model. The graph shows the price difference C(S, V, �) − cBS (S, V, �). We are taking � = 0.10 and � = 0. This example was used in the original Heston paper.

Heston Prices via Mixing and Vol-of-Vol Series 0.06 0.04 Price Difference

0.02 0 ‒0.02 ‒0.04 ‒0.06 ‒0.08 Mixing Vol-of-Vol Series

‒0.1 ‒0.12 70

80

90

100

110

120

130

Stock (USD)

FIGURE 1.3 Comparing the volatility-of-volatility series expansion with the Monte-Carlo mixing model. The graph shows the price difference C(S, V, �) − cBS (S, V, �). We are taking � = 0.20 and � = 0. This example was used in the original Heston paper.

42

INSIDE VOLATILITY FILTERING

LOCAL VOLATILITY STOCHASTIC VOLATILITY MODELS As previously mentioned, one known issue with SV models such as Heston (or other) is that given their parametric form they typically cannot match options markets exactly, which would create residual PNL. LV, on the other hand, matches options market prices exactly by construction, however does not have the proper dynamics for many cases such as forward skew or. . . One possible compromise as suggested by [216] is to have a stochastic volatility model with a superimposed local volatility component. We will use the same notations as [216] here. In general, having a stock process dS = (r − q)dt + �(S, t)Z(t)SdWS (t) with, for example d ln Z = �(�(t) − ln Z)dt + �dWZ (t), we can view the Z(t) term as the stochastic volatility piece giving the desired dynamics albeit partially, the �(S, t)Z(t) term, the local volatility (without the expectations as per below), and the �(S, t) term the residual. More precisely, as we had already seen, the local variance is 2 �LV (K, T) = � 2 (K, T)E[Z2 (T)|S(T) = K]

and separately we know that 2 �LV (K, T)

=

�C �T

+ (r − q) �C + qC �K 1 �2 C 2 �K2

;

therefore, having the local volatility we can get our residual piece via � 2 (K, T) =

2 (K, T) �LV . E[Z2 (T)|S(T) = K]

And writing this E[Z2 (T)|S(T) = K] = Ψ(K, T) we can calculate it as ∫0 Z2 p(K, Z, T)dZ ∞

Ψ(K, T) =

∫0 p(K, Z, T)dZ ∞

with p(S, V, T), the forward joint transition density of S and Z.

43

The Volatility Problem

This density can then be determined via the forward Kolmogorov equation sequentially as the authors suggest or alternatively via a particle ilter algorithm in a Monte-Carlo simulation. The calibration of the model therefore includes the usual off-line SV part to estimate �, �, �(t) via least-squares and the on-line part via the aforementioned sequence for the local volatility piece.

STOCHASTIC IMPLIED VOLATILITY As we have already mentioned, a few authors [45, 59, 224] try to model the Black-Scholes implied volatility (instead of the instantaneous variance) as a stochastic variable. In what follows in this section, we will use the notations of [45]. Assuming zero rates, borrow, and dividends, we can write for the spot process dSt = St �t dWt where �t is the instantaneous volatility38 and is stochastic. Calling �t the stochastic implied volatility for strike K and maturity T we have the price of a call option at time t as Ct = C(t, T, K) = �(St , �t (T, K), T − t, K) where � is the usual Black Scholes pricing function �(S, �, �, K) = S (h1 ) − K (h2 ) with the usual h1 =

log KS + 21 � 2 � √ � �

log KS − 21 � 2 � h2 = , √ � � and we assume the implied volatility satisies the SDE d�t = mt (T, K, St , �t , �t )dt + vt (T, K, St , �t , �t )dZt where mt and vt are the drift and volatility process for the implied volatility. 38

The square-root of our usual instantaneous variance.

44

INSIDE VOLATILITY FILTERING

Other than the usual boundedness and positivity constraints, we have the key constraint reducing the degrees of freedom �t (t, St ) = �t . Indeed, for a zero time to maturity, the implied volatility process should converge toward the instantaneous volatility. As shown in [45] one can see that the call option Ct follows the SDE dCt =  (h1 )St �t dWt +

√ ′ T − t (h1 )vt dZt [ √ ′ �t2 + T − t (h1 ) mt + 2�t (T − t) ] �t h1 h2 |vt |2 h2 �t vt . − √ − + 2(T − t) 2�t � T−t t

The traded option price (under zero rates) should be a martingale, giving the condition mt =

√ 1 [�t2 − �t2 − (T − t)h1 h2 |vt |2 + 2 T − th2 �t vt ] 2�t (T − t)

which will lead to dCt =  (h1 )St �t dWt +

√ ′ T − tSt  (h1 )vt dZt .

Now assuming vt = �t ut , we effectively have 1 d�t = 2�t (T − t)

( �t2 + 1∕4�t4 (T − t)2 |ut |2 − �t2 (T − t)�t ut

) | K ||2 | −|�t + ut log | dt + �t ut dZt St | |

Based on an alternate formulation, [45] shows there is existence and uniqueness of the solution to this system of equations.

45

The Volatility Problem

The SDE for the implied volatility is d�t = h(�t , vt , ut )dt + �t ut dZt with the constraint:

√ vt = �(t; t, St )

and dynamics [ 1 d�t = 2(T − t)�t +

(

√ K �t2 − vt + ut ln St

)2 ] dt

1 (T − t)�t3 u2t dt + �t ut dZt 8

and for example the vol of implied vol ut follows some SDE such as dut = f (ut )dt + ΥdZ⊥t where Υ is a model parameter. We will use this model later to estimate the dynamics of the instantaneous volatility from historic option prices. As we will see, the quality of observations for historic option prices (or equivalently implied volatilities) are superior to the ones for the stock prices.

JOINT SPX AND VIX DYNAMICS In order to have an SV model representing not only SPX but also the VIX dynamics, we need more than one factor of stochasticity. VIX index was created by CBOE and futures and options on VIX have increased in trading volume over the past ten years tremendously. The actual deinition of VIX is based on SPX puts and calls as explained in http://www.cboe.com/micro/VIX/vixintro.aspx but one can represent the (nontraded) VIX spot in an abstract way by √ Vt =

Q

Et

[

∫t

]

t+�

vu du

where � always corresponds to 30 calendar days, and vt is the instantaneous variance of SPX.

46

INSIDE VOLATILITY FILTERING

Following the same notations, we can write VIX futures (with maturity T) as Q FtT = Et [VT ], and in the same manner they deine options on futures with same maturity T. There have been several models suggested in literature. A popular one is the Bergomi [35] model. We are using the same notations as the author in this section. The idea is to have a two-factor SV model with abstract OU processes dXt = k1 Xt dt + dWtX dYt = k1 Yt dt + dWtY with ⟨dW X , dW Y ⟩ = �dt, ⟨dW X , dW⟩ = �SX dt, ⟨dW Y , dW⟩ = �SY dt where dW is the Wiener process for the equity (SPX). We then take the linear combination xTt = �� [(1 − �)e−k1 (T−t) Xt + �e−k2 (T−t) Yt ] √ with �� a normalization factor equal to 1∕ (1 − �)2 + � 2 + 2��(1 − �). Deining the forward variance as the stochastic variable [ �tT = E

∫t

]

T

vu du|t ,

we set �tT = �0T f T (xTt , t) where �0T is the variance swap level to maturity T, and f T is a function to be determined parametrized for each maturity T. Note that Xt , Yt , and therefore xTt are by construction drift-less.39 The local martingale condition (zero drift) on �tT provides the Feynman-Kac PDE �f T � 2 (T − t) � 2 f T =0 + �t 2 �x2 39 It is important (as the author mentions) to note that unlike in interest rates markets, in equities one cannot take one function for St = f (Wt , t) since we typically have multiple expirations Ti for contracts on the same traded asset S.

47

The Volatility Problem

with � 2 (�) = ��2 [(1 − �)2 e−2k1 � + � 2 e−2k2 � + 2��(1 − �)e−(k1 +k2 )� ]. In practice, we would choose one f T per VIX future maturity interval [Ti , Ti+1 ]. The author then suggests a weighted exponential form for f T such as f T (x, T) = (1 − �T )e�T x e−

�2 h(t,T) T 2

+ �T e�T �T x e−

� 2 �2 h(t,T) T T 2

with h(t, T) = ∫T−t � 2 (�)d�. Note that the instantaneous variance of the equity spot is simply �tt , so the risk-neutral dynamics are T

√ dS = (r − q)Sdt + S �tt dWt . Note that in practice, working with discrete maturities Ti for VIX futures, we can deine a function �i (S, t) with ( dS = (r − q)Sdt + S�i

Ti+1

1 S , STi Ti+1 − Ti ∫Ti

) �TTi dT

dWt

which will make the calculations less cumbersome.

PURE-JUMP MODELS Variance Gamma An alternative point of view is to drop the diffusion assumption altogether and replace it with a pure-jump process. Note that this is different from the jump-diffusion process previously discussed. Madan et al. suggested the following framework called Variance-Gamma (VG) in [192]. We would have the log-normal-like stock process d ln St = (�S + �)dt + X(dt; �, �, �) where, as before, �S is the real-world statistical drift of the stock log-return and � = 1� ln(1 − �� − � 2 �∕2).

48

INSIDE VOLATILITY FILTERING

As for X(dt; �, �, �) it has the following meaning: X(dt; �, �, �) = B(�(dt, 1, �); �, �) where B(dt; �, �) would be a Brownian Motion with drift � and volatility �. In other words √ B(dt; �, �) = �dt + � dtN(0, 1), and N(0, 1) is a standard Gaussian realization. The time-interval at which the Brownian Motion is considered is not dt but �(dt, 1, �), which is a random realization following a Gamma distribution with a mean 1 and variance-rate �. The corresponding probability density function is f� (dt, �) =

�

dt −1 �

dt

��Γ

�

e− � ( ) dt �

where Γ(x) is the usual Gamma function. Note that the stock log-return density could actually be integrated for the VG model, and the density of ln(St ∕S0 ) is known and could be implemented via K� (x) the modiied Bessel function of the second kind. Indeed, calling z = ln(Sk ∕Sk−1 ) and h = tk − tk−1 and posing xh = z − �S h − h� ln(1 − �� − � 2 �∕2) we have ( ) h −1 2� 4 x2h 2 exp(�xh ∕� 2 ) p(z|h) = h √ ( ) 2� 2 ∕� + � 2 � � 2��Γ h� ) ( √ 1 2 2 2 × Kh−1 xh (2� ∕� + � ) . �2 � 2 What is more, as Madan et al. show, the option valuation under VG is fairly straightforward and admits an analytically tractable closed-form which can be implemented via the above modiied Bessel function of second kind and a degenerate hypergeometric function. All details are available in [192]. Remark on the Gamma Distribution The Gamma Cumulative Distribution Function (CDF) could be deined as x

P(a, x) =

1 e−t ta−1 dt. Γ(a) ∫0

49

The Volatility Problem

Note that with our notations F� (h, x) = F(h, x, � = 1, �) with F(h, x, �, �) =

( Γ

1 �2 h �

)

( ) �2 h x �t �2 h � � e− � t � −1 dt. ∫0 �

In other words ( F(h, x, �, �) = P

) �2 h �x , . � �

The behavior if this CDF is displayed in Figure 1.4 for different values of the parameter a > 0 and for 0 < x < +∞. Using the inverse of this CDF we can have a simulated data-set for the Gamma law: x(i) = F�−1 (h, U (i) [0, 1])

with 1 ≤ i ≤ Nsims and U (i) [0, 1] a uniform random realization between zero and one. The Gamma Cumulative Distribution Function 1

P(a,x)

0.8

0.6

0.4 a = 10 a=3 a=1 a = 0.5

0.2

0

0

200

400

600

800 100x

1000

1200

1400

1600

FIGURE 1.4 The Gamma Cumulative Distribution Function P(a, x) for various values of the parameter a. The implementation is based on code available in “Numerical Recipes in C.”

50

INSIDE VOLATILITY FILTERING

Stochastic Volatility vs. Time-Changed Processes As mentioned in [24], this alternative formulation leading to time-changed processes, is closely related to the previously discussed stochastic volatility approach in the following way: Taking the above VG stochastic differential equation √ d ln St = (�S + �)dt + ��(dt, 1, �) + � �(dt, 1, �)N(0, 1), one could consider � 2 �(t, 1, �) as the integrated-variance and deine vt (�) the instantaneous-variance as � 2 �(dt, 1, �) = vt (�)dt in which case, we would have √ d ln St = (�S + �)dt + (�∕� 2 )vt (�)dt + vt (�)dtN(0, 1) √ = (�S + � + (�∕� 2 )vt (�))dt + vt (�)dZt where dZt is a Brownian Motion. This last equation is a traditional stochastic volatility one. Modified Bessel Function of Second Kind 3.5 K (x, nu = 0.1) 3

K(x, nu = 0.1)

2.5 2 1.5 1 0.5 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

The Modiied Bessel Function of Second Kind for a given parameter. The implementation is based on code available in “Numerical Recipes in C.”

51

The Volatility Problem Modified Bessel Function of Second Kind 0.95 K (x = 0.5, nu)

K(x = 0.5, nu)

0.945 0.94 0.935 0.93 0.925 0.92

0

0.05

0.1 nu

0.15

0.2

The Modiied Bessel Function of Second Kind as a function of the parameter. The implementation is based on code available in “Numerical Recipes in C.”

Variance Gamma with Stochastic Arrival An extension to the VG model would be a Variance Gamma model with Stochastic Arrival (VGSA), which would include the volatility clustering effect. This phenomenon (also represented by GARCH) means that a high (low) volatility will be followed by a series of high (low) volatilities. In this approach, we replace the dt in the previously deined f� (dt, �) with yt dt where yt follows a Square-Root (CIR) process √ dyt = �(� − yt )dt + � yt dWt where the Brownian Motion dWt is independent from other processes in the model. This is therefore a VG process where the arrival time itself is stochastic. The mean-reversion of the Square-Root process will cause the volatility persistence effect that is empirically observed. Note that (not counting �S ) the new model parameter-set is Ψ = (�, �, �, �, �, �). Option Pricing under VGSA The option pricing could be carried out via a Monte-Carlo simulation algorithm under the risk-neutral measure, where, as before, �S is replaced with r − q. We irst would simulate the path of yt by writing √ √ yk = yk−1 + �(� − yk−1 )Δt + � yk−1 ΔtZk

52

INSIDE VOLATILITY FILTERING

then calculate

N−1

YT =

∑

yk Δt

k=0

and inally apply one-step simulations T ∗ = F�−1 (YT , U[0, 1]) and40

√ ln ST = ln S0 + (r − q + �)T + �T ∗ + � T ∗ Bk .

Note that we have two normal random-variables Bk , Zk as well as a Gamma-distributed random variable T ∗ , and that they are all uncorrelated. Once the stock price ST is properly simulated, we can calculate the option price as usual. The Characteristic Function As previously discussed, another way to tackle the option-pricing issue would be to use the characteristic functions. For VG, the characteristic function is ) �2 t

( Ψ(u, t) = E[eiuX(t) ] =

1 1 − i �� u

�

.

Therefore, the log-characteristic function could be written as �(u, t) = ln(Ψ(u, t)) = t�(u, 1). In other words, E[eiuX(t) ] = Ψ(u, t) = exp(t�(u, 1)). Using which, the VGSA characteristic function becomes E[eiuX(Y(t)) ] = E[exp(Y(t)�(u, 1))] = �(−i�(u, 1)) with �() the CIR characteristic function, namely �(ut ) = E[exp(iuYt )] = A(t, u) exp(B(t, u)y0 ) 40

This means that T in VG, is replaced with YT . The rest remains identical.

53

The Volatility Problem

where exp(� 2 �t∕�2 )

A(t, u) =

2��

[cosh(�t∕2) + �∕� sinh(�t∕2)] �2 B(t, u) =

2iu � + � coth(�t∕2)

and

√

�=

� 2 − 2�2 iu.

This allows us to determine the VGSA characteristic function, which we can use to calculate options prices via numeric Fourier inversion as described in [52] and [55].

Variance Gamma with Gamma Arrival Rate For the Variance Gamma with Gamma Arrival Rate (VGG), as before, the stock process under the risk-neutral framework is d ln St = (r − q + �)dt + X(h(dt); �, �, �) with � =

1 �

ln(1 − �� − � 2 �∕2) and X(h(dt); �, �, �) = B(�(h(dt), 1, �); �, �)

and the general Gamma Cumulative Distribution Function for �(h, �, �) is ( ) �2 h x �t �2 h � � e− � t � −1 dt F(�, �; h, x) = ( 2 ) � h ∫0 � Γ 1

�

and here h(dt) = dYt with Yt is also Gamma-distributed dYt = �(dt, �a , �a ). The parameter-set is therefore Ψ = (�a , �a , �, �, �).

CHAPTER

2

The Inference Problem

In applying option pricing models, one always encounters the dificulty that the spot volatility and the structural parameters are unobservable. —Gurdip Bakshi, Charles Cao & Zhiwu Chen

INTRODUCTION Regardless of which speciic model we are using, it seems that we cannot avoid the issue of calibration. There are two possible sets of data that we can use for estimating the model parameters: options prices and historic stock prices.1 Using options prices via a least square estimator (LSE) has the obvious advantage of guaranteeing that we will match the used option market prices within a certain tolerance. However, the availability of option data is typically limited, which would force us to use interpolation and extrapolation methods. These data manipulation approaches might deteriorate the quality and the smoothness of our inputs. More importantly, matching a set of plain-vanilla option prices does not necessarily mean that we would obtain the correct price for another more exotic derivative.

1

Recently some researchers have also tried to use Historic Option Prices. See for instance Elliott [98] or Van der Merwe [240]. Inside Volatility Filtering: Secrets of the Skew, Second Edition. Alireza Javaheri © 2015 by Alireza Javaheri. Published 2015 by John Wiley & Sons Inc.

55

56

INSIDE VOLATILITY FILTERING

Using stock prices has the disadvantage of offering no guarantee of matching option prices. However supposing that the model is right, we do have a great quantity of data-input for calibration, which is a powerful argument in favor of this approach. It is important however to note that in using historic stock prices we are assuming that our time-step Δt is small enough so that we are almost in a continuous setting. What is more, we are assuming the validity of the Girsanov theorem, which is applicable to a diffusion-based model. This also means we are implicitly assuming that the market price of volatility risk is stable and so are the risk-neutral volatility-drift parameters. More accurately, having for instance a real-world model p

dvt = (� − �vt )dt + �vt dZt with p = 0.5 corresponding to the Heston model, we know that the risk-neutral volatility-drift parameter is p−1

� (r) = � + ��vt

.

As a result, supposing that � (r) is a stable (or even constant) parameter is equivalent to supposing that �, the market-price-of-volatility-risk2 , veriies 1−p

� = �vt

with � a constant coeficient. The implication of this assumption for a model with a real-world parameter-set Ψ = (�, �, �, �) and a risk-neutral counterpart Ψ(r) = (�(r) , � (r) , � (r) , �(r) ) is � = � (r) � = �(r) � = �(r) � = � (r) − �

2

p

Note that many call the market-price-of-volatility-risk the quantity ��vt .

57

The Inference Problem

Let us insist on the fact that the previous assumption3 is valid only for a diffusion-based model. For some non-Gaussian pure-jump models such as VGG, we lose the comparability between the statistical and the risk-neutral parameters. We could instead use the stock price time-series to determine the statistical density p(z) on the one hand, and use the options-prices to determine the risk-neutral density q(z) on the other; and calculate the ratio r(z) = p(z)∕q(z) corresponding to the Radon-Nikodym derivative of the two measures for this model. The central question of this chapter is therefore the Inference of the parameters embedded in a stochastic volatility model. The summary of the logical subdivisions of the problem is as follows: 1. Cross-Sectional vs. Time-Series: The former uses options prices at a given point in time and the latter a series of the underlying prices for a given period. As mentioned earlier, the former provides an estimation of the parameters in the risk-neutral universe and the latter estimation takes place in the statistical universe. 2. Classical vs. Bayesian: Using Time-Series, one could suppose that there exists an unknown but ixed set of parameters and try to estimate them as closely as possible. This is a Classical (frequentist) approach. Alternatively one could use a Bayesian approach where the parameters are supposed to be random variables and have their prior distributions that one can update via the observations.

3

Also as stated by Bakshi, Cao & Chen [21]: When the risk-aversion coeficient of the representative agent is bounded within a reasonable range, the parameters of the true distributions will not differ signiicantly from their risk-neutral counterparts. The direct implication of this is � ≈ � (r)

More importantly, for daily data we have Δt = o

(√ ) Δt

and therefore using either the real-world asset-drift �S or the dividend-adjusted risk-free rate r − q would not make a difference in parameter estimation. Some [11] even ignore the stock drift term altogether.

58

INSIDE VOLATILITY FILTERING

3. Learning vs. Likelihood-Maximization: Under the Classical hypothesis, one could either try to estimate the instantaneous variance together with the ixed parameters, which corresponds to a Learning process. A more robust way would be to estimate the Likelihood function and maximize it over all the possible values of the parameters. 4. Gaussian vs. Non-Gaussian: In any of the earlier approaches, the stochastic volatility (SV) model could be diffusion-based or not. As we will see further, this will affect the actual estimation methodology. Among the Gaussian SV models we consider are Heston, GARCH, and 3/2. Among the non-Gaussian ones are Bates, VGSA, and VGG. 5. State-Space Representation: For each of the previous approaches and for each SV model, we have a number of ways of choosing a state and representing the instantaneous variance as well as the spot price. Needless to say, a more parsimonious and lower-dimension state is preferable. 6. Diagnostics & Sampling Distribution: Once the inference process is inished, one has to verify its accuracy via various tests. Quantities such as MPE, RMSE, Box-Ljung, or � 2 numbers correspond to some of the possible diagnostics. Observing the sampling distribution over various paths is another way of checking the validity of the inference methodology. Finally, it is worth noting that our entire approach is based on Parametric stochastic volatility models. This model class is more restrictive than the non- or semi-parametric ones; however, it has the advantage of offering the possibility of a direct interpretation of the resulting parameters.

USING OPTION PRICES Using a set of current vanilla option prices, we can perform an LSE and assess the risk-neutral model parameters. Taking a set of J strike prices Kj ’s with their corresponding option prices Cmkt (Kj ) for a given maturity, we would try to minimize J ∑ (Cmodel (Kj ) − Cmkt (Kj ))2 . j=1

The minimization4 could, for example, be done via the Direction Set (Powell) method, the Conjugate Gradient (Fletcher-Reeves-Polak-Ribiere) 4 Some consider that this minimization will give more importance to the ATM options and try therefore to correct this by introducing weights in the summation. There are also Entropy-based techniques as discussed in [17] applied to local volatility models, which are different from our parametric models.

59

The Inference Problem

method or the Levenberg-Marquardt (LM) method. We will now briely describe the Powell optimization algorithm. We will give some detail in what follows on these optimization algorithms.

Conjugate Gradient (Fletcher-Reeves-Polak-Ribiere) Method The idea is to minimize a function f (x) by inding the point where gradient is zero. This essentially follows a steepest descent algorithm. Given a function, its gradient ∇x f indicates the direction of maximum increase. To minimize, one can start in the opposite direction and perform a line-search to reach the minimum of the function. So Δx0 = −∇x f (x0 ) �0 = argmin f (x0 + �Δx0 ) �

x1 = x0 + �0 Δx0 and continue iteration n = 0, 1, . . . in the same manner Δxn = −∇x f (xn ) �n =

Δxtn (Δxn − Δxn−1 ) . Δxtn Δxn

New conjugate direction is sn = Δxn + �n sn−1 . The line minimization will give �0 = argmin f (xn + �sn ) �

and inally xn+1 = xn + �n sn to close the loop, and stop when a desired precision is reached.

Levenberg-Marquardt (LM) Method This optimization methodology is also known as the damped least-squares method. It essentially interpolates between the Gauss Newton method

60

INSIDE VOLATILITY FILTERING

and the gradient descent algorithm. Assuming we are trying to minimize a sum m ∑ [yi − f (xi , �)]2 S(�) = i=1

with respect to a parameter set �, we proceed as follows. We start with an initial guess for the parameter set � and at each iteration of the optimization, we replace it with � + �. In order to determine � we write f (xi , � + �) ≈ f (xi , �) + Ji � with Ji =

�f (xi , �) ��

the gradient of f with respect to �. We can therefore say S(� + �) ≈

m ∑

[yi − f (xi , �) − Ji �]2 .

i=1

We then can take the derivative with respect to � and set the result to zero by using vector notations (Jt J)� = Jt (y − f (�)) where J is the Jacobian matrix of gradients. The Levenberg algorithm replaces this with the “damped” version (Jt J + �I)� = Jt (y − f (�)) with I the identity matrix. Later, Marquardt replaced this identity matrix with the diagonal matrix consisting of the diagonal elements of Jt J, resulting in the Levenberg Marquardt algorithm: (Jt J + �diag(Jt J))� = Jt (y − f (�))

61

The Inference Problem

Direction Set (Powell) Method The optimization method we will use later is the Direction Set (Powell) method and does not require any Gradient or Hessian computation.5 This is a quadratically convergent method producing mutually conjugate directions. Most multidimensional optimization algorithms require a onedimensional line minimization routine, which does the following: Given as input the vectors P and n and the function f , ind the scalar � that minimizes f (P + �n), then replace P with P + �n and n with �n. The idea would be to take a set of directions that are as noninterfering as possible in order to avoid spoiling one minimization with the subsequent one. This way an interminable cycling through the set of directions will not occur. This is why we seek conjugate directions. Calling the function to be minimized f (x) with x a vector of dimension N, we can write the second-order Taylor expansion around a particular point P f (x) ≈ f (P) + ∇f (P)x + where Hij =

�2 f �xi �xj

1 xHx 2

is the Hessian matrix of the function at point P.

We therefore have for the variation of the gradient �(∇f ) = H�x and in order to have a noninterfering new direction, we choose v such as the motion along v remains perpendicular to our previous direction u u�(∇f ) = uHv = 0. In this case, the directions u and v are said to be conjugate. Powell suggests a quadratically convergent method that produces a set of N mutually conjugate directions. The following description is taken from Press [214] where the corresponding source code could be found as well. 1. Initialize the set of directions ui to the basis vectors for i = 1, . . . , N. 2. Save the starting point as P0 . 5 This is an important feature when the function to be optimized contains discontinuities.

62

INSIDE VOLATILITY FILTERING

3. Move Pi−1 to the minimum along direction ui and call it Pi . 4. Set ui to ui+1 for i = 1, . . . , N − 1 and set uN to PN − P0 . 5. Move PN to the minimum along uN and call this point P0 and go back to step 2. For a quadratic form, k iterations of this algorithm will produce a set of directions whose last k members are mutually conjugate. The idea is to repeat the steps until the function stops decreasing. However, this procedure tends to produce directions that are linearly dependent and therefore provides us with the minimum only over a sub-space. Hence, the idea of discarding the direction along which f made the largest decrease. This seems paradoxical because we are dropping our best direction in the new iteration; however, this is the best chance of avoiding a buildup of linear dependence. In what follows we apply the Powell algorithm to SPX options valued via the mixing Monte-Carlo method.

Numeric Tests We apply the Powell algorithm to SPX options valued via the Mixing Monte-Carlo method. The optimization is performed across close-tothe-money strike prices as of t0 = May 5, 2002, with the index S0 = 1079.88 and maturities T = August 17, 2002, T = September 21, 2002, T = December 21, 2002, and T = March 22, 2003 (see Figures 2.1 to 2.5). Implied Volatility

Volatility Surface

0.4 0.35 0.3 0.25 0.2 0.15 0.1

700

800 900 1000 1100 Strike 1200 1300 Price 1400 1500 0.2

0.8 0.7 0.6 rity 0.5 atu 0.4 to M 0.3 ime T

0.9

S&P500 Volatility Surface as of May 21, 2002, with Index = 1079.88 USD. The surface will be used for itting via the Direction Set (Powell) Optimization algorithm applied to a Square-Root model implemented with a one-factor Monte-Carlo mixing method.

63

The Inference Problem Volatility Smile Fitting 300 Market 08/17/2002 Model 08/17/2002

250

Call Price

200 150 100 50 0 800

850

900

950 1000 1050 1100 1150 1200 1250 1300 Strike Price

Mixing Monte-Carlo simulation with the Square-Root model for SPX on May 21, 2002, with Index = 1079.88 USD, maturity, August 17, 2002. Powell (Direction Set) Optimization method was used for Least-Square Calibration Optimal Parameters �̂ = 0.081575, �̂ = 3.308023, �̂ = 0.268151, �̂ = −0.999999. Volatility Smile Fitting 350 Market 09/21/2002 Model 09/21/2002

300

Call Price

250 200 150 100 50 0 700

800

900

1000

1100

1200

1300

1400

Strike Price

Mixing Monte-Carlo simulation with the Square-Root model for SPX on May 21, 2002, with Index = 1079.88 USD, maturity, September 21, 2002. Powell (Direction Set) Optimization method was used for Least-Square Calibration Optimal Parameters �̂ = 0.108359, �̂ = 3.798900, �̂ = 0.242820, �̂ = −0.999830.

64

INSIDE VOLATILITY FILTERING Volatility Smile Fitting 400 Market 12/21/2002 Model 12/21/2002

350

Call Price

300 250 200 150 100 50 0 700

800

900

1000

1100

1200

1300

1400

1500

Strike Price

Mixing Monte-Carlo simulation with the Square-Root model for SPX on May 21, 2002, with Index = 1079.88 USD, maturity, December 21, 2002. Powell (Direction Set) Optimization method was used for Least-Square Calibration Optimal Parameters �̂ = 0.126519, �̂ = 3.473910, �̂ = 0.222532, �̂ = −0.999991. Volatility Smile Fitting 400 Market 03/22/2002 Model 03/22/2002

350

Call Price

300 250 200 150 100 50 0 700

800

900

1000

1100

1200

1300

1400

1500

Strike Price

Mixing Monte-Carlo simulation with the Square-Root model for SPX on May 21, 2002, with Index = 1079.88 USD, maturity, March 22, 2003. Powell (Direction Set) Optimization method was used for Least-Square Calibration Optimal Parameters �̂ = 0.138687, �̂ = 3.497779, �̂ = 0.180010, �̂ = −1.000000.

65

The Inference Problem

TABLE 2.1 The estimation is performed for SPX on t = May 21, 2002, with Index = 1079.88 USD for different maturities T. ̂ �

�̂

�̂

0.081575 0.108359 0.126519 0.138687

3.308023 3.798900 3.473910 3.497779

0.268151 0.242820 0.222532 0.180010

T 8/17/2002 9/21/2002 12/21/2002 3/22/2003

�̂ −0.999999 −0.999830 −0.999991 −1.000000

As we see in Table 2.1, the estimated parameters are fairly stable for different maturities and therefore the stochastic volatility model seems to be fairly time-homogeneous.

The Distribution of the Errors Since the parameter-set Ψ contains only a few elements and we can have many options prices, it is clear that the matching of the model and market prices is not perfect. Hence, the idea of observing the distribution of the errors { } ̂ exp − 1 Υ2 + Υ (j) (0, 1) Cmkt (Kj ) = Cmodel (Kj , Ψ) 2

̂ the optimal with 1 ≤ j ≤ J and Υ the error standard-deviation, and Ψ parameter-set. As usual,  (0, 1) is the standard normal distribution. Note that our previously discussed LSE approach is not exactly equivalent to the maximization of a Likelihood function based on the above distribution, since the latter would correspond to the minimization of the sum of the squared log returns. A good bias test would be to check for the predictability of the errors. For this, one could run a regression of the error ̂ ej = Cmkt (Kj ) − Cmodel (Kj , Ψ) on a few factors corresponding for instance to moneyness or maturity. A low R2 for the regression would prove that the model errors are not predictable and there is no major bias. For a detailed study, see [192] for instance.

USING STOCK PRICES The Likelihood Function If as in the previous section we use European options with a given maturity T and with different strike prices, then we will be estimating q(ST |S0 ; Ψ)

66

INSIDE VOLATILITY FILTERING

which corresponds to the risk-neutral density, given a known current stock price S0 and given a constant parameter-set Ψ. As discussed, the least-squares estimation (LSE) is used to ind the best guess for the unknown ideal parameter-set. On the other hand if we use a time series of stock prices (St )0≤t≤T , we would be dealing with the joint probability p(S1 , . . . , ST |S0 ; Ψ) which we can rewrite as p(S1 , . . . , ST |S0 ; Ψ) =

T ∏

p(St |St−1 , . . . , S0 ; Ψ).

t=1

It is this joint probability that is commonly referred to as the Likelihood Function L0∶T (Ψ). Maximizing the Likelihood over the parameter-set Ψ would provide us with the best parameter-set for the statistical density p(ST |S0 ; Ψ). Note that we are using a Classical (Frequentist) approach where we assume that the parameters are unknown but ixed over [0, T]. In other words, we would be dealing with the same parameter-set for any of the p(St |St−1 , . . . , S0 ; Ψ) with 1 ≤ t ≤ T. It is often convenient to work with the log of the Likelihood function since this will produce a sum

ln L0∶T (Ψ) =

T ∑

ln p(St |St−1 , . . . , S0 ; Ψ).

t=1

The Justification for the MLE As explained for instance in [105], one justiication of the maximization of the (log) Likelihood function comes from the Kullback-Leibler (KL) distance. The KL distance is deined as6 d(p∗ , p) =

∫

p∗ (x)(ln p∗ (x) − ln p(x))dx

where p∗ (x) is the ideal density, while p(x) the density under estimation.

6

Hereafter, when the bounds are not speciied, the integral is taken on the entire space of the integrand argument.

67

The Inference Problem

We can write d(p∗ , p) = E∗ ln(p∗ (x)∕p(x)). Note that using the Jensen (log convexity) inequality d(p∗ , p) = −E∗ ln(p(x)∕p∗ (x)) ≥ − ln(E∗ (p(x)∕p∗ (x))) so

d(p∗ , p) ≥ − ln

∫

p∗ (x)p(x)∕p∗ (x)dx = 0

and d(p, p∗ ) = 0 if and only if p = p∗ , which conirms that d(.,.) is a distance. Now minimizing d(p, p∗ ) over p() would be equivalent to minimizing the term − p∗ (x) ln p(x)dx ∫ since the rest of the expression depends on p∗ () only. This latter expression could be written in the discrete framework, having T observations S1 , . . . , ST as −

T ∑

ln p(St )

t=1

since the observations are by assumption distributed according to the ideal p∗ (). This justiies our maximizing T ∏

p(St ).

t=1

Note that in a pure parameter estimation, this would be the MLE approach. However, the minimization of the KL distance is more general and can allow for Model Identiication. MLE has many desirable asymptotic attributes as explained for example in [134]. Indeed ML estimators are consistent and converge to the right parameter-set as the number of observations increases. They actually reach the lower bound for the error, referred to as the Cramer-Rao bound, which corresponds to the inverse of the Fisher Information matrix. Calling the irst derivative of the log-likelihood the score function h(Ψ) =

� ln L0∶T (Ψ) , �Ψ

68

INSIDE VOLATILITY FILTERING

it is known that MLE could be interpreted as a special case of the General Method of Moments (GMM) where the moment g(Ψ) such that E[g(Ψ)] = 0 is simply taken to be the above score function. Indeed, we would then have E[h(Ψ)] = which means that

∫

� ln L0∶T (Ψ) L0∶T (Ψ)dz0∶T = 0 �Ψ

∫

�L0∶T (Ψ) dz0∶T = 0 �Ψ

as previously discussed in the MLE. Note that taking the derivative of the above with respect to the parameter-set (using one-dimensional notations for simplicity) � (h(Ψ)L0∶T (Ψ))dz0∶T = 0 ∫ �Ψ which will give us �L0∶T (Ψ)

� 2 ln L0∶T (Ψ) �L0∶T (Ψ) �Ψ L0∶T (Ψ)dz0∶T = − dz0∶T 2 ∫ ∫ �Ψ L0∶T (Ψ) �Ψ ( )2 � ln L0∶T (Ψ) L0∶T (Ψ)dz0∶T =− ∫ �Ψ meaning that [( )2 ] ] � 2 ln L0∶T (Ψ) � ln L0∶T (Ψ)  = −E =E �Ψ2 �Ψ [

which is referred to as the Information Matrix Identity. As previously stated, asymptotically, we have for the optimal ̂ and the ideal Ψ∗ parameter-set Ψ ̂ − Ψ∗ ∼  (0,  −1 ). Ψ Likelihood Evaluation and Filtering For GARCH models the Likelihood is known under an integrated form. Indeed, calling ut the mean-adjusted

69

The Inference Problem

stock return, vt the variance, and (Bt ) a Gaussian sequence, we have for any GARCH model ut = h(vt , Bt ) and vt = f (vt−1 , ut−1 ; Ψ) where f () and h() are two deterministic functions. This will allow us to directly determine and optimize7 L1∶T (Ψ) ∝ −

T ∑

ln(vt ) +

t=1

u2t . vt

This is possible because GARCH models have one source of randomness and there is a time-shift between the variance and the spot equations. Unlike GARCH, most stochastic volatility models have two (nonperfectly correlated) sources of randomness (Bt ) and (Zt ) and have equations of the form ut = h(vt , Bt ) vt = f (vt−1 , Zt ; Ψ) which means that the Likelihood function is not directly known under an integrated form, and we need iltering techniques for its estimation and optimization. Another justiication for iltering is its application to Parameter Learning. As we shall see, in this approach we use the joint distribution of the hidden state and the parameters. In order to obtain the optimal value of the hidden state vt given all the observations z1∶t , we need to use a ilter.

Filtering The idea here is to use the Filtering theory for the estimation of stochastic volatility model parameters. What we are trying to do is to ind the probability density function (pdf) corresponding to a state xk at time step k given all the observations z1∶k up to that time. 7

We generally drop constant terms in the likelihood function since they do not affect the optimal arguments. Hence the notation L1∶T (Ψ) ∝ . . .

70

INSIDE VOLATILITY FILTERING

Looking for the pdf p(xk |z1∶k ) we can proceed in two stages: 1. First, we can write the Time Update iteration by applying the Chapman-Kolmogorov equation8 p(xk |z1∶k−1 ) =

∫

p(xk |xk−1 , z1∶k−1 )p(xk−1 |z1∶k−1 )dxk−1

=

∫

p(xk |xk−1 )p(xk−1 |z1∶k−1 )dxk−1

by using the Markov property. 2. Following this, for the Measurement Update we use the Bayes rule p(xk |z1∶k ) =

p(zk |xk )p(xk |z1∶k−1 ) p(zk |z1∶k−1 )

where the denominator p(zk |z1∶k−1 ) could be written as p(zk |z1∶k−1 ) =

∫

p(zk |xk )p(xk |z1∶k−1 )dxk

and corresponds to the Likelihood function for the time-step k. Proof Indeed, we have p(xk |z1∶k ) =

p(z1∶k |xk )p(xk ) p(z1∶k )

=

p(zk , z1∶k−1 |xk )p(xk ) p(zk , z1∶k−1 )

=

p(zk |z1∶k−1 , xk )p(z1∶k−1 |xk )p(xk ) p(zk |z1∶k−1 )p(z1∶k−1 )

=

p(zk |z1∶k−1 , xk )p(xk |z1∶k−1 )p(z1∶k−1 )p(xk ) p(zk |z1∶k−1 )p(z1∶k−1 )p(xk )

=

p(zk |xk )p(xk |z1∶k−1 ) p(zk |z1∶k−1 )

which proves the claimed result. (QED) 8

See Shreve [229], for instance.

71

The Inference Problem

Note that we use the fact that at time-step k the value of z1∶k is perfectly known. The Kalman ilter (detailed next) is a special case where the distributions are normal and could be written as p(xk |zk−1 ) =  (x̂ −k , P−k ) p(xk |zk ) =  (x̂ k , Pk )

Indeed, in the special Gaussian case, each distribution could be entirely characterized via its irst two moments. However, it is important to remember that the Kalman ilter (KF) is optimal in the Gaussian linear case. In the nonlinear case, it will indeed always be sub-optimal. Interpretation of the Kalman Gain The basic idea behind the KF is the following observation. Having x a normally distributed random-variable with a mean mx and variance Sxx , and z a normally distributed random-variable with a mean mz and variance Szz , and having Szx = Sxz the covariance between x and z, the conditional distribution of x|z is also normal with mx|z = mx + K(z − mz ) with K = Sxz S−1 zz . Interpreting x as the hidden-state and z as the observation, the matrix K would correspond to the Kalman ilter in the linear case. We also have Sx|z = Sxx − KSxz . An alternative interpretation of the Kalman ilter could be based on linear regression. Indeed, if we knew the time-series of (zk ) and (xk ), then the regression could be written as xk = �zk + � + �k with � the slope, � the intercept, and (�k ) the residuals. It is known that under a least squares regression, we have � = Sxz S−1 zz which again is the expression for the Kalman gain. We now will describe various nonlinear extensions of the KF.

72

INSIDE VOLATILITY FILTERING

The Simple and Extended Kalman Filters The irst algorithms we choose here are the simple and extended Kalman ilters9 due to their well-known lexibility and ease of implementation. The simple or traditional Kalman ilter (KF) applies to linear Gaussian cases, while the extended KF (EKF) could be used for nonlinear Gaussian cases via a irst-order linearization. We shall therefore describe EKF and consider the simple KF as a special case. In order to clarify the notations, let us briely rewrite the EKF equations. Given a dynamic process xk following a possibly nonlinear transition equation (2.1) xk = f(xk−1 , wk ), we suppose we have a measurement zk via a possibly nonlinear observation equation (2.2) zk = h(xk , uk ) where wk and uk are two mutually uncorrelated sequences of temporally uncorrelated normal random-variables with zero means and covariance matrices Qk , Rk respectively.10 Moreover, wk is uncorrelated with xk−1 and uk uncorrelated with xk . We deine the linear a priori process estimate as x̂ −k = E[xk ]

(2.3)

which is the estimation at time step k − 1 prior to measurement. Similarly, we deine the linear a posteriori estimate x̂ k = E[xk |zk ]

(2.4)

which is the estimation at time step k after the measurement. We also have the corresponding estimation errors e−k = xk − x̂ −k and ek = xk − x̂ k and the estimate error covariances t

P−k = E[e−k e−k ] Pk =

E[ek etk ]

(2.5) (2.6)

where the superscript t corresponds to the transpose operator. 9

For a description see for instance Welch [245] or Harvey [136]. Some prefer to write xk = f(xk−1 , wk−1 ). Needless to say, the two notations are equivalent.

10

73

The Inference Problem

We now deine the Jacobian matrices of f with respect to the system process and the system noise as Ak and Wk respectively. Similarly, we deine the gradient matrices of h with respect to the system process and the measurement noise as Hk and Uk respectively. More accurately, for every row i and column j we have Aij = �fi ∕�xj (x̂ k−1 , 0)

Wij = �fi ∕�wj (x̂ k−1 , 0)

Hij = �hi ∕�xj (x̂ −k , 0)

Uij = �hi ∕�uj (x̂ −k , 0)

We therefore have the following Time Update equations: x̂ −k = f(x̂ k−1 , 0)

(2.7)

P−k = Ak Pk−1 Atk + Wk Qk−1 Wtk .

(2.8)

and

We deine the Kalman gain as the matrix Kk used in the Measurement Update equations (2.9) x̂ k = x̂ −k + Kk (zk − h(x̂ −k , 0)) and Pk = (I − Kk Hk )P−k

(2.10)

where I represents the Identity matrix. The optimal Kalman gain corresponds to the mean of the conditional distribution of xk upon the observation zk or equivalently, the matrix that would minimize the mean square error Pk within the class of linear estimators. This optimal gain is Kk = P−k Htk (Hk P−k Htk + Uk Rk Utk )−1

(2.11)

The previous equations complete the KF algorithm. Another Interpretation of the Kalman Gain Note that an easy way to observe that Kk minimizes the a posteriori error covariance Pk is to consider the one-dimensional linear case x̂ k = x̂ −k + Kk (zk − Hk x̂ −k ) = x̂ −k + Kk (zk − Hk xk + Hk e−k ) so ek = xk − x̂ k = e−k − Kk (uk + Hk e−k ).

74

INSIDE VOLATILITY FILTERING

Therefore Pk = E(e2k ) = P−k + K2k (Rk + Hk2 P−k + 0) − 2Kk Hk P−k and taking the derivative with respect to Kk and setting it to zero, we get Kk =

P−k Hk Hk2 P−k + Rk

which is the one-dimensional expression for the linear Kalman gain. Residuals, MPE, and RMSE In what follows we shall call the estimated observations ẑ −k . For the simple and extended Kalman ilters, we have ẑ −k = h(x̂ −k , 0). The residuals are the observation errors, deined as z̃ k = zk − ẑ −k . Needless to say, the smaller these residuals, the higher the quality of the ilter. Therefore, to measure the performance, we deine the Mean Price Error (MPE) and Root Mean Square Error (RMSE) as the mean and standard-deviation of the residuals 1 ∑ z̃ MPE = N k=1 k √ √ N √1 ∑ RMSE = √ (̃z − MPE)2 . N k=1 k N

The Unscented Kalman Filter Recently, Julier and Uhlmann [174] proposed a new extension of the Kalman ilter to nonlinear systems, completely different from the EKF. They argue that EKF could be dificult to implement and more importantly dificult to tune and only reliable for systems that are almost linear within the update intervals. The new method called the unscented Kalman ilter (UKF) will calculate the mean to a higher order of accuracy than the EKF and the covariance to the same order of accuracy. Unlike the EKF, this method does not require any Jacobian calculation since it does not approximate the nonlinear functions of the process and the

75

The Inference Problem

observation. Therefore, it uses the true nonlinear models but approximates the distribution of the state random variable xk by applying an Unscented Transformation to it. As we will see next, we construct a set of Sigma Points, which capture the mean and covariance of the original distribution and when propagated through the true nonlinear system, capture the posterior mean and covariance accurately to the third order. Similarly to the EKF, we start with an initial choice for the state vector x̂ 0 = E[x0 ] and its Covariance Matrix P0 = E[(x0 − x̂ 0 )(x0 − x̂ 0 )t ]. We then concatenate the space vector with the system noise and the observation noise11 and create an augmented state vector for each step k greater than one ⎛ xk−1 ⎞ xak−1 = ⎜wk−1 ⎟ ⎜ ⎟ ⎝ uk−1 ⎠ and therefore

⎛x̂ k−1 ⎞ x̂ ak−1 = ⎜ � ⎟ ⎜ ⎟ ⎝ � ⎠

and

Pak−1

⎞ ⎛ Pxw (k − 1|k − 1) � Pk−1 ⎟ ⎜ = ⎜Pxw (k − 1|k − 1) Pww (k − 1|k − 1) � ⎟ ⎜ � � Puu (k − 1|k − 1)⎟⎠ ⎝

for each iteration k. The augmented state will therefore have a dimension na = nx + nw + nu . We then need to calculate the corresponding Sigma Points through the Unscented Transformation: �ak−1 (0) = x̂ ak−1 . For i = 1, . . . , na �ak−1 (i) = x̂ ak−1 +

11

(√

(na + �)Pak−1

) i

This space augmentation will not be necessary if we have additive noises as in xk = f (xk−1 ) + wk−1 and zk = h(xk ) + uk .

76

INSIDE VOLATILITY FILTERING

and for i = na + 1, . . . , 2na �ak−1 (i) = x̂ ak−1 −

) (√ (na + �)Pak−1

i−na

where the subscripts i and i − na correspond to the ith and i − nth a columns of the square-root matrix.12 This prepares us for the Time Update and the Measurement Update equations, similarly to the EKF. The Time Update equations are �k|k−1 (i) = f(�xk−1 (i), �w k−1 (i)) for i = 0, . . . , 2na + 1 and x̂ −k =

2na ∑

Wi(m) �k|k−1 (i)

i=0

and P−k =

2na ∑

Wi(c) (�k|k−1 (i) − x̂ −k )(�k|k−1 (i) − x̂ −k )t

i=0

where the superscripts x and w respectively correspond to the state and system-noise portions of the augmented state. The Wi(m) and Wi(c) weights are deined as W0(m) = and W0(c) =

� na + �

� + (1 − � 2 + �) na + �

and for i = 1, . . . , 2na Wi(m) = Wi(c) =

12

1 . 2(na + �)

The square-root matrix is calculated via Singular Value Decomposition (SVD) and Cholesky factorization [214]. In case Pak−1 is not Positive-Deinite, one could for example use a truncation procedure.

77

The Inference Problem

The scaling parameters �, �, �, and � = � 2 (na + �) − na will be chosen for tuning. We also deine within the Time Update equations Zk|k−1 (i) = h(�k|k−1 (i), �uk−1 (i)) and ẑ −k =

2na ∑

Wi(m) Zk|k−1 (i)

i=0

where the superscript u corresponds to the observation-noise portion of the augmented state. As for the Measurement Update equations, we have Pzk zk = and Pxk zk =

2na ∑

Wi(c) (Zk|k−1 (i) − ẑ −k )(Zk|k−1 (i) − ẑ −k )t

i=0

2na ∑

Wi(c) (�k|k−1 (i) − x̂ −k )(Zk|k−1 (i) − ẑ −k )t

i=0

which gives us the Kalman gain Kk = Pxk zk P−1 zk zk and we get as before x̂ k = x̂ −k + Kk (zk − ẑ −k ) where again zk is the observation at time (iteration) k. Also, we have Pk = P−k − Kk Pzk zk Ktk which completes the Measurement Update equations.

Kushner’s Nonlinear Filter It would be instructive to compare this algorithm to the nonlinear iltering algorithm based on an approximation of the conditional distribution by Kushner et al. [182]. In this approach, the authors suggest using a Gaussian

78

INSIDE VOLATILITY FILTERING

Quadrature in order to calculate the integral at the measurement update (or the time update) step.13 As the Kushner paper indicates, having an N-dimensional normal random-variable X =  (m, P) with m and P the corresponding mean and covariance, for a polynomial G of degree 2M − 1 we can write14 ] (y − m)t P−1 (y − m) E[G(X)] = dy G(y) exp − N 1 ∫ 2 (2�) 2 |P| 2 [

1

which is equal to E[G(X)] =

M ∑ i1 =1

...

M ∑

wi1 . . . wiN G(m +

√

P� )

iN =1

) ( where � t = �i1 , . . . , �iN is the vector of the Gauss-Hermite roots of order M and wi1 , . . . , wiN are the corresponding weights. Note that even if both Kushner’s NLF and UKF use Gaussian Quadratures, UKF only uses 2N + 1 sigma points, while NLF needs MN points for the integral computation. Kushner and Budhiraja suggest using this technique primarily for the measurement update (iltering) step. They claim that provided this step is properly implemented, the time update (prediction) step can be carried out via a linearization similarly to the EKF. Details of the Kushner Algorithm Let us use the same notations as for the UKF algorithm. We therefore have the augmented state xak−1 and its covariance Pak−1 as before. Here, for a Quadrature order of M on an N-dimensional variable, the sigma-points are deined for j = 1, . . . , N and ij = 1, . . . , M as �ak−1 (i1 , . . . , iN ) = x̂ ak−1 +

√ Pak−1 � (i1 , . . . , iN )

where the square-root here corresponds to the Cholesky factorization, and again � (i1 , . . . , iN )[j] = �ij for each j between 1 and the dimension N and each ij between 1 and the Quadrature order M.

13

The analogy between Kushner’s nonlinear ilter and the unscented Kalman ilter has already been studied in Ito & Xiong [158]. 14 A description of the Gaussian Quadrature could be found in Press et al. [214].

79

The Inference Problem

Similarly to the UKF, we have the Time Update equations �k|k−1 (i1 , . . . , iN ) = f(�xk−1 (i1 , . . . , iN ), �w k−1 (i1 , . . . , iN )) but now x̂ −k =

M ∑

...

M ∑

wi1 . . . wiN �k|k−1 (i1 , . . . , iN )

iN =1

i1 =1

and P−k =

M ∑

...

i1 =1

M ∑

wi1. . .wiN (�k|k−1 (i1 , . . . , iN ) − x̂ −k )(�k|k−1 (i1 , . . . , iN ) − x̂ −k )t .

iN =1

Again, we have Zk|k−1 (i1 , . . . , iN ) = h(�k|k−1 (i1 , . . . , iN ), �uk−1 (i1 , . . . , iN )) and ẑ −k =

M ∑ i1 =1

...

M ∑

wi1 . . . wiN Zk|k−1 (i1 , . . . , iN ).

iN =1

Therefore, the Measurement Update equations will be Pzk zk =

M M ∑ ∑ wi1. . .wiN (Zk|k−1 (i1 ,. . ., iN ) − ẑ −k )(Zk|k−1 (i1 ,. . ., iN ) − ẑ −k )t ... i1 =1

iN =1

and M M ∑ ∑ Pxk zk = . . . wi1. . .wiN (�k|k−1 (i1 ,. . ., iN ) − x̂ −k )(Zk|k−1 (i1 ,. . ., iN ) − ẑ −k )t i1 =1

iN =1

which gives us the Kalman gain Kk = Pxk zk P−1 zk zk and we get as before x̂ k = x̂ −k + Kk (zk − ẑ −k ) where again zk is the observation at time (iteration) k.

80

INSIDE VOLATILITY FILTERING

Also, we have Pk = P−k − Kk Pzk zk Ktk which completes the Measurement Update equations. When N = 1 and � = 2, the numeric integration in the UKF will correspond to a Gauss-Hermite Quadrature of order M = 3. However, in the UKF we can tune the ilter and reduce the higher term errors via the previously mentioned parameters � and �. Note that when h(x, u) is strongly nonlinear, the Gauss Hermite integration is not eficient for evaluating the moments of the measurement update equation, since the term p(zk |xk ) contains the exponent zk − h(xk , uk ). The iterative methods based on the idea of importance sampling proposed in [182] correct this problem at the price of a strong increase in computation time. As suggested in [158], one way to avoid this integration would be to make the additional hypothesis that xk , h(xk , uk )|z1∶k−1 is Gaussian.

Parameter Learning One important issue to realize is that the Kalman ilter can be used either for State Estimation (Filtering) or for Parameter Estimation (Machine Learning). When we have both State Estimation and Parameter Estimation we are dealing with a dual or joint estimation. This latter case is the one concerning us since we are estimating the state volatility as well as the model parameters. As explained in Haykin’s book [140], in a dual ilter (DF) we separate the state vector from the parameters and we apply two intertwined ilters to them. On the other hand, in a joint ilter (JF), we concatenate the state vector and the parameters and apply one ilter to this augmented state. Note that in the dual ilter we need to compute recurrent derivatives with respect to parameters, while in a joint ilter no such step is needed. It is possible to interpret the JF in the following way. In a regular ilter, that is, iltering of the state xk for a ixed set of parameters Ψ0 , we are maximizing the conditional density p(x1∶k |z1∶k , Ψ0 ) and as we said, to do that we write p(x1∶k |z1∶k , Ψ0 ) =

p(z1∶k |x1∶k , Ψ0 )p(x1∶k |Ψ0 ) p(z1∶k |Ψ0 )

so we maximize the above with respect to the state xk for a given set of parameters.

81

The Inference Problem

This means that the optimal state x̂ 1∶k for a given parameter-set is given by x̂ 1∶k = argmax[p(z1∶k , x1∶k |Ψ0 )]. As we will see, in an MLE approach we use this optimal state iltering for each iteration of the likelihood-maximization over the parameter-set Ψ. In a joint ilter, on the other hand, we are directly optimizing the joint conditional density p(x1∶k , Ψ|z1∶k ) which we can also write as p(x1∶k , Ψ|z1∶k ) =

p(z1∶k |x1∶k , Ψ)p(x1∶k |Ψ)p(Ψ) . p(z1∶k )

Given that the denominator is functionally independent of x1∶k and Ψ, and given that p(Ψ) contains no prior information,15 the maximization will be on p(z1∶k |x1∶k , Ψ)p(x1∶k |Ψ) = p(z1∶k , x1∶k |Ψ). ̂ are In other words in a JF, the optimal state x̂ 1∶k and parameter-set Ψ found by writing ̂ = argmax[p(z1∶k , x1∶k |Ψ)]. (x̂ 1∶k , Ψ) In what follows, we apply the joint EKF methodology to a few examples. An Illustration Before using this technique for the SV model, let us take a simple example where �k = �k−1 + � + 0.10wk and

zk = �k + 0.10uk

where � ≈ 3.14159 and wk , uk are independent Gaussian random variables. The linear state-space system could be written as ( ) ( ( ) ) �k wk 1 1 xk = + 0.10 = x �k 0 0 1 k−1 and

15

( ) zk = 1 0 xk + 0.10uk

Again we are in a Frequentist framework, not Bayesian.

82

INSIDE VOLATILITY FILTERING Joint Filter

3.4 3.2 3 2.8 2.6 2.4 2.2 pi[k] pi

2 1.8

0

2

4

6

8

10

12

14

Observations

FIGURE 2.1 A simple example for the joint ilter. The convergence toward the constant parameter � happens after a few iterations. We choose the initial values �0 = z0 = 0 and �0 = 1.0. We also take Q = 0.1I2 and R = 0.10. Applying the Kalman ilter to an artiicially generated data-set, we plot the resulting �k in Figure 2.1. As we can see, the parameter converges very quickly to its true value. Even if we associated a noise of 0.10 to the constant parameter � we can see that for 5000 observations, taking the mean of the iltered state between observations 20 and 5000 we get �̂ = 3.141390488 which is very close to the value 3.14159 used in data generation process. Joint Filtering Examples After going through this simple example, we now apply the JF technique to our stochastic volatility problem. We shall study a few examples in order to ind the best state space representation. Example 1 Our irst example would be the Square-Root SV model: ( ) √ √ 1 ln Sk = ln Sk−1 + �S − vk−1 Δt + vk−1 ΔtBk−1 2 √ √ vk = vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk−1 To simplify, we suppose that the value of �S is known.

83

The Inference Problem

We can now deine the state variable16 ⎛ln Sk ⎞ ⎜ vk ⎟ ⎜ � ⎟ ⎟ xk = ⎜ ⎜ � ⎟ ⎜ � ⎟ ⎜ ⎟ ⎝ � ⎠ (

and the system noise wk = with its covariance matrix

Bk Zk

)

( ) 1 � Qk = � 1

and therefore

) ( √ √ 1 ⎛ln S Δt + + � − v ΔtBk−1 ⎞⎟ v k−1 S k−1 k−1 2 ⎜ √ √ ⎜ v ⎟ ⎜ k−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk−1 ⎟ ⎟ � xk = f(xk−1 , wk−1 ) = ⎜ ⎜ ⎟ � ⎜ ⎟ � ⎜ ⎟ ⎜ ⎟ � ⎠ ⎝

and therefore the Jacobian Ak is ⎛1 − 1 Δt 0 0 2 ⎜ ⎜0 1 − �Δt Δt −vk−1 Δt ⎜0 0 1 0 Ak = ⎜ 0 0 0 1 ⎜ ⎜0 0 0 0 ⎜0 0 0 0 ⎝ and

16

0 0 0 0 1 0

0⎞ ⎟ 0⎟ 0⎟ ⎟ 0⎟ 0⎟ 1⎟⎠

√ √ ⎛ vk−1 Δt ⎞ 0 √ ⎟ ⎜ √ 0 � vk−1 Δt⎟ ⎜ ⎜ ⎟ 0 0 Wk = ⎜ ⎟. 0 0 ⎜ ⎟ ⎜ ⎟ 0 0 ⎜ ⎟ 0 0 ⎝ ⎠

In reality, we should write the estimation parameters �k , �k , �k , and �k . However, we drop the indexes for simplifying the notations.

84

INSIDE VOLATILITY FILTERING

Having the measurement zk = ln Sk , we can write ( ) Hk = 1 0 0 0 0 0 and Uk = 0. We could, however, introduce a measurement noise R corresponding to the intraday stock price movements and the bid-ask spread, in which case we would have zk = ln Sk + R�k where �k represents a sequence of uncorrelated standard normal random variables. This means that Rk = R and Uk = 1. We can then tune the value of R in order to get more stable results. Example 2 The same exact methodology could be used in the GARCH framework. We deine the state variable xtk = (ln Sk , vk , �0 , �, �, c) and take for observation the logarithm of the actual stock price Sk . The system could be written as xk = f(xk−1 , wk−1 ) with wk = Bk a one-dimensional source of noise with a variance Qk = 1 and ) √ ( 1 ⎛ln S + vk−1 Bk−1 ⎞⎟ + � − v k−1 S 2 k−1 ⎜ )2 ⎟ ( ⎜ √ ⎟ ⎜ �0 + �vk−1 + � Bk−1 − c vk−1 ⎟ ⎜ f(xk−1 , wk−1 ) = ⎜ �0 ⎟ ⎟ ⎜ � ⎟ ⎜ � ⎟ ⎜ ⎟ ⎜ c ⎠ ⎝ and the Jacobian ⎛1 − 21 ⎜ 2 ⎜0 � + �c ⎜0 0 Ak = ⎜ 0 ⎜0 ⎜0 0 ⎜ 0 ⎝0

0

0

0

1 1 0 0 0

c2 vk−1

vk−1 0 0 1 0

0 1 0 0

⎞ ⎟ 2�cvk−1 ⎟ 0 ⎟ ⎟ 0 ⎟ 0 ⎟ ⎟ 1 ⎠ 0

85

The Inference Problem

and

√ ⎛ vk−1 ⎞ ⎟ ⎜ √ ⎜−2�c vk−1 ⎟ ⎟ ⎜ 0 ⎟. Wk = ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎠ ⎝ The observation zk will be zk = h(xk ) = ln(Sk )

exactly as in the previous example. The rest of the algorithm would therefore be identical to the one included in Example 1. Example 3 In Example 2, we included all the variables in the system process and we observed part of the system. It is also possible to separate the measurement and the system variables as follows. Taking a general discrete stochastic volatility process as17 ( ) √ √ 1 ln Sk = ln Sk−1 + �S − vk Δt + vk ΔtBk 2 √ vk = vk−1 + b(vk−1 )Δt + a(vk−1 ) ΔtZk with Bk and Zk two normal random sequences with a mean of zero and variance one, with √ a correlation equal to �. Posing yk = vk Zk and performing the usual Cholesky factorization √ Bk = �Zk + 1 − �2 Xk where Zk and Xk are uncorrelated, we can now take the case of a Square-Root process and write ⎛vk ⎞ ⎜y ⎟ ⎜ k⎟ ⎜�⎟ xk = ⎜ ⎟ ⎜�⎟ ⎜�⎟ ⎜ ⎟ ⎝�⎠ 17

Note that the indexing here is slightly different from the previous examples.

86

INSIDE VOLATILITY FILTERING

and xk = f(xk−1 , Zk ) with √ ⎛ vk−1 + (� − �vk−1 )Δt + � √vk−1 ΔtZk ⎞ ⎟ ⎜ √ 1 √ ⎟ ⎜ 2 ⎜(vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk ) Zk ⎟ ⎟ ⎜ � + QZk ⎟ f(xk−1 , Zk ) = ⎜ ⎟ ⎜ � + QZk ⎟ ⎜ ⎟ ⎜ � + QZ k ⎟ ⎜ ⎟ ⎜ � + QZk ⎠ ⎝ which provides us with the Jacobian ⎛1 − �Δt ⎜ 0 ⎜ ⎜ 0 Ak = ⎜ ⎜ 0 ⎜ 0 ⎜ ⎝ 0 and

0 Δt −vk−1 Δt 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0

0 0 0 0 1 0

0⎞ 0⎟⎟ 0⎟ 0⎟⎟ 0⎟ ⎟ 1⎠

√ √ ⎞ ⎛ � vk−1 Δt ⎟ ⎜ 1 ⎜(v + (� − �vk−1 )Δt) 2 ⎟ k−1 ⎟ ⎜ Q ⎟ Wk = ⎜ ⎟ ⎜ Q ⎟ ⎜ Q ⎟ ⎜ ⎟ ⎜ Q ⎠ ⎝ The measurement equation is ( ) ( ) √ √ √ √ Sk 1 = �S − vk Δt + � Δtyk + 1 − �2 vk ΔtXk zk = ln Sk−1 2

and therefore

) ( √ Hk = − 1 Δt � Δt 0 0 0 0 2

√ √ √ with uk = Xk and Uk = 1 − �2 vk Δt, which completes our set of equations. Again, we could tune the system noise Q in order to obtain more stable results.

87

The Inference Problem Extended Kalman Filter 4 Historic Parameter

3.5 3 Omega

2.5 2 1.5 1 0.5 0

0

200

400

600

800

1000

1200

1400

Days

EKF estimation (Example 1) for the drift parameter � SPX Index daily close prices were used over ive years from October 1, 1996, to October 28, 2001. The convergence is fairly good.

Extended Kalman Filter 14 Historic Parameter

12

Theta

10 8 6 4 2 0

0

200

400

600

800

1000

1200

1400

Days

EKF estimation (Example 1) for the drift parameter � SPX Index daily close prices were used over ive years from October 1, 1996, to September 28, 2001. The convergence is fairly good.

88

INSIDE VOLATILITY FILTERING Extended Kalman Filter 0.18 Historic Parameter

0.16 0.14

XI

0.12 0.1 0.08 0.06 0.04 0.02 0

0

200

400

600

800

1000

1200

1400

Days

EKF estimation (Example 1) for the volatility-of-volatility parameter � SPX Index daily close prices were used over ive years from October 1, 1996, to September 28, 2001. The convergence is rather poor. We shall explain this via the concept of observability. Extended Kalman Filter

0.1

Historic Parameter

0.05

Rho

0 –0.05 –0.1 –0.15 –0.2 –0.25

0

200

400

600

800

1000

1200

1400

Days

EKF estimation (Example 1) for the correlation parameter � SPX Index daily close prices were used over ive years from October 1, 1996, to September 28, 2001. The convergence is rather poor. We shall explain this via the concept of observability.

89

The Inference Problem

Observability From the previous tests, it seems that the EKF provides us with a nonrobust calibration methodology. Indeed, the results are very sensitive to the choice of system noise Q and observation noise R. We chose for this case Q = 10−3 and R ≈ 0. This brings to attention the issue of observability. A nonlinear system with a state vector xk of dimension n is observable if ⎛ H ⎞ ⎜ HA ⎟ ⎟ ⎜ O = ⎜ HA2 ⎟ ⎜ ... ⎟ ⎟ ⎜ ⎝HAn−1 ⎠ has a full rank of n. For an explanation refer to Reif et al. [215]. It is fairly easy to see that among the aforementioned examples, the irst and third one (corresponding to the SV formulation) have for the observation matrix O a rank of four and therefore are not observable. This explains why they do not converge well and are so sensitive to the tuning parameters Q and R. This means that the choices of the state variables for the Examples 1 and 3 were rather poor. One reason is that in our state space choice, we considered zk = h(vk−1 , . . . ) and which implies that

xk = (. . . , vk , . . . ) = f (xk−1 , . . . ) �h = 0. �vk

We shall see how to correct this in the next section by choosing a more appropriate state space representation. The One-Dimensional State within the Joint Filter Considering the state equation √ √ vk = vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk−1 ( [ ] ) √ √ 1 − �� ln Sk−1 + �S − vk−1 Δt + vk−1 ΔtBk−1 − ln Sk 2 posing for every k

̃k = √ 1 (Zk − �Bk ), Z 1 − �2

90

INSIDE VOLATILITY FILTERING

̃ k uncorrelated with Bk . Therefore, considering we will have as expected Z the augmented state ⎛vk ⎞ ⎜ ⎟ ⎜�⎟ xk = ⎜ � ⎟ ⎜ ⎟ ⎜�⎟ ⎜�⎟ ⎝ ⎠ we will have the state transition equation ] [ ( ) ⎛v ⎞ + (� − ���S ) − � − 21 �� vk−1 Δt k−1 ⎜ ⎟ ( ) √ √ √ ⎜ ⎟ Sk 2 ̃ ⎜ +�� ln Sk−1 + � 1 − � vk−1 ΔtZk−1 ⎟ ⎟ ̃ k−1 ) = ⎜⎜ � f(xk−1 , Z ⎟ ⎜ ⎟ � ⎜ ⎟ � ⎜ ⎟ ⎜ ⎟ � ⎝ ⎠ and the measurement equation would be ( ) √ √ 1 zk = ln Sk+1 = ln Sk + �S − vk Δt + vk ΔtBk . 2 The corresponding EKF Jacobians for this system are ) ) ( ( ( ( S S ⎛ ⎞ � ln S k � ln S k ) ( ⎜ ) )⎟ ( k−1 ( k−1 1 ⎜1 − � − 2 �� Δt −vk−1 Δt − �S − 12 vk−1 − �S − 12 vk−1 ⎟ ⎜ ×Δt ⎟ ) ) ⎜ ⎟ × Δt × Δt ⎟ Ak = ⎜ ⎜ ⎟ 0 1 0 0 0 ⎜ ⎟ 0 0 1 0 0 ⎜ ⎟ ⎜ ⎟ 0 0 0 1 0 ⎜ ⎟ 0 0 0 0 1 ⎝ ⎠ √ √ ⎞ ⎛� 1 − �2 √v k−1 Δt ⎟ ⎜ 0 ⎟ ⎜ ⎟ Wk = ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎠ ⎝

91

The Inference Problem

) ( Hk = − 12 Δt 0 0 0 0 √ √ Uk = vk Δt It is easy to check that this system is observable since the observation matrix Ok is of full rank. This shows that our state space choice is better than the previous ones. The UKF would be implemented in a similar fashion from the transition and observation equations above. Again, for the UKF we would not need to compute any Jacobians. An important issue to consider is that of tuning. We could add extra noise to the observation and hence lower the weight associated with the observations. In this case after choosing a tuning parameter R we would write ) (√ √ Uk = vk Δt R and Uk Utk = vk Δt + R2 . The choice of the initial conditions and the tuning parameters could make the algorithm fail or succeed. It therefore seems that there is little robustness in this procedure. We consider the example of 5000 artiicial data points artiicially produced via a Heston stochastic volatility process with a parameter-set Ψ∗ = (� = 0.10, � = 10.0, � = 0.03, � = −0.50) with a given �S = 0.025. We then choose a tuning parameter R = 0.10 and take a reasonable guess for the initial-conditions Ψ0 = (�0 = 0.15, �0 = 10.0, �0 = 0.02, �0 = −0.51) and apply the joint ilter. The results are displayed in Figures 2.2 to 2.5. As we can see, the convergence for � and � are better than the two others. We shall see later why this is. Allowing a burn-in period of 1000 points, we can calculate the mean (and the standard-deviation) of the generated parameters after the simulation 1000. Joint Filters and Time Interval One dificulty with the application of the JF to the stochastic volatility problem, is the following: Unless we are √ dealing √ with a longer time-interval such as Δt = 1, the observation error vk ΔtBk

92

INSIDE VOLATILITY FILTERING

Joint EKF 0.9 0.8 0.7 0.6 0.5 0.4 omega 0.3

0

500

1000

1500

2000

2500

Days

FIGURE 2.2 Joint EKF estimation for the parameter � Prices were simulated with Ψ∗ = (0.10, 10.0, 0.03, −0.50). The convergence remains mediocre. We shall explain this in the following section.

Joint EKF 10.005 theta

10 9.995 9.99 9.985 9.98 9.975 9.97 9.965

0

500

1000

1500

2000

2500

Days

FIGURE 2.3 Joint EKF estimation for the parameter � Prices were simulated with Ψ∗ = (0.10, 10.0, 0.03, −0.50). The convergence remains mediocre. We shall explain this in the following section.

93

The Inference Problem

Joint EKF 0.09 0.08

xi

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 ‒0.01 0

500

1000

1500

2000

2500

Days

FIGURE 2.4 Joint EKF estimation for the parameter � Prices were simulated with Ψ∗ = (0.10, 10.0, 0.03, −0.50). The convergence remains mediocre. We shall explain this in the following section.

Joint EKF ‒0.494 ‒0.495 ‒0.496 ‒0.497 ‒0.498 ‒0.499 ‒0.5 ‒0.501

rho

‒0.502 0

500

1000

1500

2000

2500

Days

FIGURE 2.5 Joint EKF estimation for the parameter � Prices were simulated with Ψ∗ = (0.10, 10.0, 0.03, −0.50). The convergence remains mediocre. We shall explain this in the following section.

94

INSIDE VOLATILITY FILTERING

is too large compared to the sensitivity of the ilter with respect to the state through −0.5vk Δt. Indeed for a Δt = 1∕252 we have18 √ Δt = o( Δt). A simple Monte-Carlo test will allow us to verify this: We simulate a Heston model on the one hand, and on the other a modiied model where we multiply both Brownian Motions by a factor Δt. This will make the errors smaller by a factor of 252 for the daily case. We call this the modiied model. After generating 5000 data points with a parameter set (�∗ = 0.10, � ∗ = 10.0, � ∗ = 0.03, �∗ = −0.50) and a drift �S = 0.025, we suppose we know all parameters except �. We then apply the JKF to ind the estimate �. ̂ We can observe in Figure 2.6 that the ilter diverges when applied to the Heston model but converges fast when applied to the modiied model. However, in reality we have no control over the observation error, which is precisely the volatility! 0.5 0.4 0.3 0.2 0.1 0 ‒0.1 ‒0.2 0.10 JKF / Heston JKF / Modified SV

‒0.3 ‒0.4 ‒0.5 0

50

100

150

200

FIGURE 2.6 Joint EKF estimation for the parameter � applied to the Heston model as well as to a modiied model where the noise is reduced by a factor 252. As we can see, the convergence for the modiied model is improved dramatically. This justiies our comments on large observation error.

18

Hereafter, xh = o(yh ) means xh ∕yh → 0 as h → 0, or more intuitively, xh is much smaller than yh for a tiny h.

95

The Inference Problem

In a way, this brings up a more fundamental issue regarding the stochastic volatility estimation problem: By deinition, volatility represents the noise of the stock process. Indeed if we had taken the spot price Sk as the observation and the variance vk as the state, we would have √ √ Sk = Sk−1 + Sk−1 �S Δt + Sk−1 vk ΔtBk . We would then have an observation function gradient H = 0 and the system would be unobservable! It is precisely because we use a Taylor second-order expansion ln(1 + R) ≈ R −

1 2 R 2

that we obtain access to vk through the observation function. However, the error remains dominant as the irst order of the expansion. Some [137] have tried ( ( )) Sk 2 ln ln ≈ ln(vk ) + ln(Δt) + ln(B2k ) Sk−1 and

� ln(B2k ) ∼ −1.27 + √  (0, 1), 2

but the latter approximation may or may not be valid depending on the problem under study.

Parameter Estimation via MLE As previously stated, one of the principal methods of estimation under the classical framework is the Maximization of the Likelihood. Indeed, this estimation method has many desirable asymptotic properties. Therefore, instead of using the ilters alone, we could separate the parameter-set Ψ = (�, �, �, �) from the state vector (ln Sk , vk ) and use the Kalman Filter for state iltering within each MLE iteration19 and estimate the parameters iteratively.

19 To be more accurate, since the noise process is conditionally Gaussian, we are dealing with a Quasi-Maximum-Likelihood (QML) Estimation. More detail could be found for instance in Gourieroux [130].

96

INSIDE VOLATILITY FILTERING

An Illustration Let us irst consider the case of the previous illustration �k = �k−1 + � + 0.10wk and

zk = �k + 0.10uk

where � ≈ 3.14159 and wk , uk are independent Gaussian random variables. Here we take xk = �k and Ak = Hk = 1 Wk = Uk = 0.1 The maximization of the Gaussian Likelihood with respect to the parameter � is equivalent to minimizing ] [ N ∑ z̃ 2k L1∶N = ln(Fk ) + Fk k=1 with residuals and

z̃ k = zk − ẑ −k = zk − x̂ −k Fk = Pzk zk = Hk P−k Hkt + Uk Rk Ukt .

Note that we used scalar notations here and in vectorial notations we would have N ∑ ̃k] [ln(|Fk |) + z̃ tk F−1 L1∶N = k z k=1

where |Fk | is the determinant of Fk . We use the scalar notations for simplicity, and also because in the stochastic volatility problem, we usually deal with one-dimensional observations (namely the stock price). The minimization via a Direction Set (Powell) method over 500 artiicially generated observation points will provide �̂ = 3.145953 very quickly.

97

The Inference Problem

Stochastic Volatility Examples For our previous irst example the system state vector now becomes ( ) ln Sk xk = vk which means the dimension of our state is now two, and ( xk = f(xk−1 , wk−1 ) =

) ( ) √ √ ln Sk−1 + �S − 21 vk−1 Δt + vk−1 ΔtBk−1 . √ √ vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk−1

The system noise is still ( wk =

Bk Zk

)

and the corresponding covariance matrix is ( Qk =

) 1 � . � 1

We have the measurement zk = ln Sk and therefore we can write ( ) Hk = 1 0 . Now for a given set of parameters (�, �, �, �) we can ilter this system with the EKF (or the UKF) using ( ) 1 − 12 Δt Ak = 0 1 − �Δt and Wk =

(√ √ vk−1 Δt 0

0

)

√ . √ � vk−1 Δt

Note that the observation matrix is ( ) 1 0 Ok = 1 − 21 Δt which is of full rank. Our system is therefore observable.

98

INSIDE VOLATILITY FILTERING

After iltering for this set of parameters, we calculate the sum to be minimized [ ] N ∑ z̃ 2k ln(Fk ) + �(�, �, �, �) = Fk k=1 with z̃ k = zk − h(x̂ −k , 0) and Fk = Hk P−k Htk + Uk Rk Utk . The minimization could once again be done via a Direction Set (Powell) method as described previously. This will avoid a calculation of the gradient ∇�. It is interesting to observe (cf. Figures 2.7 and 2.8) that the results of the EKF and UKF are very close and the ilter errors are comparable. However,

6.705 Market EKF UKF

6.7

Log Stock Price

6.695 6.69 6.685 6.68 6.675 6.67 6.665 6.66 100

105

110 Days

115

120

FIGURE 2.7 SPX historic data (1996–2001) are iltered via EKF and UKF. The ̂ = (�, ̂ �, ̂ �) results are very close; however, the estimated parameters Ψ ̂ �, ̂ differ. Indeed, we ind (�̂ = 0.073028, �̂ = 1.644488, �̂ = 0.190461, �̂ = −1.000000) for the EKF and (�̂ = 0.540715, �̂ = 13.013577, �̂ = 0.437523, �̂ = −1.000000) for the UKF. This might be due to the relative insensitivity of the ilters to the parameter-set Ψ or the non-uniqueness of the optimal parameter-set. We shall explain this low sensitivity in more detail.

99

The Inference Problem 0.0001 EKF UKF

9e-05 8e-05

Filter Error

7e-05 6e-05 5e-05 4e-05 3e-05 2e-05 1e-05 0 500

550

600

650

700

Days

FIGURE 2.8 EKF and UKF absolute iltering-errors for the same time-series. As we can see, there is no clear superiority of one algorithm over the other. the estimated parameter set Ψ = (�, �, �, �) can have a different set of values depending on which ilter is actually used.20 Hence the question, how sensitive are these ilters to Ψ? In order to answer this question we can run an estimation for EKF and use the estimated parameters in UKF and observe how good a it we obtain. The results show that this sensitivity is fairly low. Again, this might be due to the relative insensitivity of the ilters to the parameter-set Ψ or the non-uniqueness of the optimal parameter-set. As we will see, the answer to this question also depends on the sample size. Optimization-Constraints for the Square-Root Model In terms of the optimization constraints, in addition to the usual �≥0

(2.12)

�≥0 �≥0

−1 ≤ � ≤ 1

20

Note however that the values of the resulting long-term volatilities

close.

√

� �

are rather

100

INSIDE VOLATILITY FILTERING

we need to make sure that the value of the variance remains positive, that is, √ √ vk + (� − �vk )Δt + � vk ΔtZk ≥ 0

for any vk ≥ 0 and any Gaussian random value Zk . For a Gaussian random variable Zk and any positive real number Z∗ , we can write Zk ≥ −Z∗ with a probability P∗ . For instance if Z∗ = 4 then P∗ = 0.999968. Therefore, ixing a choice of Z∗ , it is almost always enough for us to have √ √ vk + (� − �vk )Δt − � vk ΔtZ∗ ≥ 0

for any vk ≥ 0. √ √ Considering the function f (x) = x + (� − �x)Δt − � x ΔtZ∗ , it is fairly easy to see that f (0) = �Δt ≥ 0 by assumption, and for x very large f (x) ≈ (1 − �Δt)x, which is positive if �≤

1 . Δt

(2.13)

This is most of the time realized for a small Δt such as ours. Now f (x) being a continuous function and having positive values at zero and ininity it would be suficient to make sure that its one minimum on [0, +∞[ is also positive. A simple derivative computation shows that xmin = � 2 Δt(Z∗ )2 and therefore the positivity is realized if21 4(1−�Δt)2 �≤

2 √ �(1 − �Δt) Z∗

(2.14)

which completes our set of constraints. An Alternative Implementation We could also perform the same estimation but based on our previous third example. Again we have ( ) √ √ 1 ln Sk = ln Sk−1 + �S − vk Δt + vk ΔtBk 2 √ √ vk = vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk with Bk and Zk two normal random sequences with a mean of zero, a variance of one, and a correlation equal to �. 21

Naturally we suppose Δt > 0.

101

The Inference Problem

However, since for a Kalman ilter the process noise and the measurement noise must be uncorrelated, we introduce yk =

√

vk Zk

and performing the usual Cholesky factorization Bk = �Zk + where Zk and Xk are uncorrelated, we can write

√ 1 − �2 Xk

( ) vk xk = yk and xk = f(xk−1 , Zk ) with √ ) √ vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk √ 1 √ (vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk ) 2 Zk

( f(xk−1 , Zk ) =

which provides us with the Jacobian ) ( 1 − �Δt 0 Ak = 0 0 (

and Wk =

)

√ √ � vk−1 Δt 1

.

(vk−1 + (� − �vk−1 )Δt) 2

The measurement equation is ( ) √ √ √ √ 1 zk = ln Sk = ln Sk−1 + �S − vk Δt + � Δtyk + 1 − �2 vk ΔtXk 2 and therefore

( √ ) Hk = − 1 Δt � Δt 2

√ √ √ with uk = Xk and Uk = 1 − �2 vk Δt which completes our set of equations. Note that the observation matrix is ( √ ) − 21 Δt � Δt Ok = − 21 Δt(1 − �Δt) 0 which is of full rank. Our system is therefore observable.

102

INSIDE VOLATILITY FILTERING

The One-Dimensional State Finally, a simpler way of writing the state-space system, which will be our method of choice hereafter, would be to subtract from both sides of the state equation xk = f (xk−1 , wk−1 ), a multiple of the quantity h(xk−1 , uk−1 ) − zk−1 which is equal to zero. This would allow us to eliminate the correlation between the system and the measurement noises. Indeed, if the system equation is ( ) √ √ 1 ln Sk = ln Sk−1 + �S − vk−1 Δt + vk−1 ΔtBk−1 2 √ √ vk = vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk−1 writing √ √ vk = vk−1 + (� − �vk−1 )Δt + � vk−1 ΔtZk−1 [ ( ) ] √ √ 1 − �� ln Sk−1 + �S − vk−1 Δt + vk−1 ΔtBk−1 − ln Sk 2 posing for every k

̃k = √ 1 Z (Zk − �Bk ), 1 − �2

̃ k uncorrelated with Bk and we will have as expected Z [ ( ) ] 1 xk = vk = vk−1 + (� − ���S ) − � − �� vk−1 Δt 2 ( ) √ √ √ Sk ̃ k−1 + �� ln + � 1 − �2 vk−1 ΔtZ Sk−1 and the measurement equation would be ( ) √ √ 1 zk = ln Sk+1 = ln Sk + �S − vk Δt + vk ΔtBk . 2

(2.15)

(2.16)

With this system, everything becomes one-dimensional and the computations become much faster both for the EKF and UKF. For the EKF we will have ( ) 1 Ak = 1 − � − �� Δt 2 and

√ √ √ Wk = � 1 − �2 vk−1 Δt

103

The Inference Problem

as well as

1 Hk = − Δt 2

and Uk =

√ √ vk Δt.

Again, for the MLE we will try to minimize �(�, �, �, �) =

N ∑ k=1

[

z̃ 2 ln(Fk ) + k Fk

]

with residuals z̃ k = zk − h(x̂ −k , 0) and Fk = Hk P−k Htk + Uk Utk . The same time update and measurement update will be used with the UKF. The ML Estimator can be used as usual. Below is a C++ routine for the implementation of the EKF applied to the Heston model: // // // // // // //

log_stock_prices are the log of stock prices muS is the real-world stock drift n_stock_prices is the number of the above stock prices (omega, theta, xi, rho) are the Heston parameters u[] is the set of means of observation errors v[] is the set of variances of observation errors estimates[] are the estimated observations from the filter

void estimate_extended_kalman_parameters_1_dim( double *log_stock_prices, double muS, int n_stock_prices, double omega, double theta, double xi, double rho, double *u, double *v, double *estimates) { int i1; double x, x1, W, H, A;

104

INSIDE VOLATILITY FILTERING

double P, P1, z, U, K; double delt=1.0/252.0; double eps=0.00001; x = 0.04; P=0.01; u[0]=u[n_stock_prices-1]=0.0; v[0]=v[n_stock_prices-1]=1.0; estimates[0]=estimates[1]=log_stock_prices[0]+eps; for (i1=1;i1